id	sid	tid	token	lemma	pos
cana-1747	1	1	communications	communication	NOUN
cana-1747	1	2	on	on	ADP
cana-1747	1	3	applied	apply	VERB
cana-1747	1	4	nonlinear	nonlinear	ADJ
cana-1747	1	5	analysis	analysis	NOUN
cana-1747	1	6	issn	issn	NOUN
cana-1747	1	7	:	:	PUNCT
cana-1747	1	8	1074	1074	NUM
cana-1747	1	9	-	-	PUNCT
cana-1747	1	10	133x	133x	NUM
cana-1747	1	11	vol	vol	NOUN
cana-1747	1	12	32	32	NUM
cana-1747	1	13	no	no	NOUN
cana-1747	1	14	.	.	NOUN
cana-1747	1	15	2	2	NUM
cana-1747	1	16	(	(	PUNCT
cana-1747	1	17	2025	2025	NUM
cana-1747	1	18	)	)	PUNCT
cana-1747	1	19	355	355	NUM
cana-1747	1	20	https://internationalpubls.com	https://internationalpubls.com	X
cana-1747	1	21	a	a	DET
cana-1747	1	22	law	law	NOUN
cana-1747	1	23	of	of	ADP
cana-1747	1	24	the	the	DET
cana-1747	1	25	iterated	iterated	ADJ
cana-1747	1	26	logarithm	logarithm	NOUN
cana-1747	1	27	for	for	ADP
cana-1747	1	28	the	the	DET
cana-1747	1	29	signum	signum	PROPN
cana-1747	1	30	function	function	PROPN
cana-1747	1	31	santosh	santosh	PROPN
cana-1747	1	32	ghimire	ghimire	PROPN
cana-1747	1	33	department	department	PROPN
cana-1747	1	34	of	of	ADP
cana-1747	1	35	applied	apply	VERB
cana-1747	1	36	sciences	science	NOUN
cana-1747	1	37	and	and	CCONJ
cana-1747	1	38	chemical	chemical	PROPN
cana-1747	1	39	engineering	engineering	NOUN
cana-1747	1	40	,	,	PUNCT
cana-1747	1	41	pulchowk	pulchowk	NOUN
cana-1747	1	42	campus	campus	PROPN
cana-1747	1	43	,	,	PUNCT
cana-1747	1	44	institute	institute	NOUN
cana-1747	1	45	of	of	ADP
cana-1747	1	46	engineering	engineering	PROPN
cana-1747	1	47	,	,	PUNCT
cana-1747	1	48	tribhuvan	tribhuvan	PROPN
cana-1747	1	49	university	university	PROPN
cana-1747	1	50	,	,	PUNCT
cana-1747	1	51	kathmandu	kathmandu	NOUN
cana-1747	1	52	,	,	PUNCT
cana-1747	1	53	nepal	nepal	ADJ
cana-1747	1	54	.	.	PUNCT
cana-1747	2	1	email	email	NOUN
cana-1747	2	2	:	:	PUNCT
cana-1747	2	3	santoshghimire@ioe.edu.np	santoshghimire@ioe.edu.np	ADJ
cana-1747	2	4	article	article	NOUN
cana-1747	2	5	history	history	NOUN
cana-1747	2	6	:	:	PUNCT
cana-1747	2	7	received	receive	VERB
cana-1747	2	8	:	:	PUNCT
cana-1747	2	9	02	02	NUM
cana-1747	2	10	-	-	SYM
cana-1747	2	11	08	08	NUM
cana-1747	2	12	-	-	PUNCT
cana-1747	2	13	2024	2024	NUM
cana-1747	2	14	revised	revise	VERB
cana-1747	2	15	:	:	PUNCT
cana-1747	2	16	12	12	NUM
cana-1747	2	17	-	-	SYM
cana-1747	2	18	09	09	NUM
cana-1747	2	19	-	-	PUNCT
cana-1747	2	20	2024	2024	NUM
cana-1747	2	21	accepted	accept	VERB
cana-1747	2	22	:	:	PUNCT
cana-1747	2	23	20	20	NUM
cana-1747	2	24	-	-	SYM
cana-1747	2	25	09	09	NUM
cana-1747	2	26	-	-	PUNCT
cana-1747	2	27	2024	2024	NUM
cana-1747	2	28	abstract	abstract	NOUN
cana-1747	2	29	in	in	ADP
cana-1747	2	30	2023	2023	NUM
cana-1747	2	31	,	,	PUNCT
cana-1747	2	32	s.	s.	PROPN
cana-1747	2	33	ghimire	ghimire	PROPN
cana-1747	2	34	established	establish	VERB
cana-1747	2	35	a	a	DET
cana-1747	2	36	one	one	NUM
cana-1747	2	37	-	-	PUNCT
cana-1747	2	38	sided	sided	ADJ
cana-1747	2	39	version	version	NOUN
cana-1747	2	40	of	of	ADP
cana-1747	2	41	a	a	DET
cana-1747	2	42	law	law	NOUN
cana-1747	2	43	of	of	ADP
cana-1747	2	44	the	the	DET
cana-1747	2	45	iterated	iterated	ADJ
cana-1747	2	46	logarithm	logarithm	NOUN
cana-1747	2	47	,	,	PUNCT
cana-1747	2	48	denoted	denote	VERB
cana-1747	2	49	as	as	ADP
cana-1747	2	50	lil	lil	PROPN
cana-1747	2	51	,	,	PUNCT
cana-1747	2	52	for	for	ADP
cana-1747	2	53	summations	summation	NOUN
cana-1747	2	54	of	of	ADP
cana-1747	2	55	signum	signum	ADJ
cana-1747	2	56	functions	function	NOUN
cana-1747	2	57	analogous	analogous	ADJ
cana-1747	2	58	to	to	ADP
cana-1747	2	59	the	the	DET
cana-1747	2	60	lil	lil	NOUN
cana-1747	2	61	proposed	propose	VERB
cana-1747	2	62	by	by	ADP
cana-1747	2	63	salem	salem	NOUN
cana-1747	2	64	and	and	CCONJ
cana-1747	2	65	zygmund	zygmund	NOUN
cana-1747	2	66	for	for	ADP
cana-1747	2	67	trigonometric	trigonometric	ADJ
cana-1747	2	68	series	series	NOUN
cana-1747	2	69	.	.	PUNCT
cana-1747	3	1	in	in	ADP
cana-1747	3	2	this	this	DET
cana-1747	3	3	article	article	NOUN
cana-1747	3	4	,	,	PUNCT
cana-1747	3	5	we	we	PRON
cana-1747	3	6	complete	complete	VERB
cana-1747	3	7	the	the	DET
cana-1747	3	8	lil	lil	NOUN
cana-1747	3	9	for	for	ADP
cana-1747	3	10	these	these	DET
cana-1747	3	11	signum	signum	ADJ
cana-1747	3	12	functions	function	NOUN
cana-1747	3	13	by	by	ADP
cana-1747	3	14	establishing	establish	VERB
cana-1747	3	15	the	the	DET
cana-1747	3	16	complimentary	complimentary	ADJ
cana-1747	3	17	version	version	NOUN
cana-1747	3	18	of	of	ADP
cana-1747	3	19	the	the	DET
cana-1747	3	20	lil	lil	PROPN
cana-1747	3	21	.	.	PUNCT
cana-1747	4	1	keywords	keyword	NOUN
cana-1747	4	2	:	:	PUNCT
cana-1747	4	3	signum	signum	PROPN
cana-1747	4	4	functions	function	NOUN
cana-1747	4	5	,	,	PUNCT
cana-1747	4	6	q	q	ADJ
cana-1747	4	7	-	-	ADJ
cana-1747	4	8	lacunary	lacunary	ADJ
cana-1747	4	9	series	series	NOUN
cana-1747	4	10	,	,	PUNCT
cana-1747	4	11	law	law	NOUN
cana-1747	4	12	of	of	ADP
cana-1747	4	13	the	the	DET
cana-1747	4	14	iterated	iterated	ADJ
cana-1747	4	15	logarithm	logarithm	NOUN
cana-1747	4	16	,	,	PUNCT
cana-1747	4	17	borelcantelli	borelcantelli	VERB
cana-1747	4	18	lemma	lemma	PROPN
cana-1747	4	19	.	.	PROPN
cana-1747	5	1	1	1	X
cana-1747	5	2	.	.	X
cana-1747	5	3	introduction	introduction	NOUN
cana-1747	5	4	the	the	DET
cana-1747	5	5	lil	lil	NOUN
cana-1747	5	6	is	be	AUX
cana-1747	5	7	a	a	DET
cana-1747	5	8	widely	widely	ADV
cana-1747	5	9	recognized	recognize	VERB
cana-1747	5	10	theorem	theorem	NOUN
cana-1747	5	11	in	in	ADP
cana-1747	5	12	probability	probability	NOUN
cana-1747	5	13	that	that	PRON
cana-1747	5	14	complements	complement	VERB
cana-1747	5	15	two	two	NUM
cana-1747	5	16	other	other	ADJ
cana-1747	5	17	fundamental	fundamental	ADJ
cana-1747	5	18	theorems	theorem	NOUN
cana-1747	5	19	:	:	PUNCT
cana-1747	5	20	the	the	DET
cana-1747	5	21	central	central	ADJ
cana-1747	5	22	limit	limit	NOUN
cana-1747	5	23	theorem	theorem	ADJ
cana-1747	5	24	(	(	PUNCT
cana-1747	5	25	clt	clt	PROPN
cana-1747	5	26	)	)	PUNCT
cana-1747	5	27	and	and	CCONJ
cana-1747	5	28	the	the	DET
cana-1747	5	29	law	law	NOUN
cana-1747	5	30	of	of	ADP
cana-1747	5	31	large	large	ADJ
cana-1747	5	32	numbers	number	NOUN
cana-1747	5	33	(	(	PUNCT
cana-1747	5	34	lln	lln	PROPN
cana-1747	5	35	)	)	PUNCT
cana-1747	5	36	.	.	PUNCT
cana-1747	6	1	while	while	SCONJ
cana-1747	6	2	the	the	DET
cana-1747	6	3	lln	lln	PROPN
cana-1747	6	4	describes	describe	VERB
cana-1747	6	5	the	the	DET
cana-1747	6	6	tendencies	tendency	NOUN
cana-1747	6	7	of	of	ADP
cana-1747	6	8	the	the	DET
cana-1747	6	9	average	average	NOUN
cana-1747	6	10	of	of	ADP
cana-1747	6	11	independent	independent	ADJ
cana-1747	6	12	random	random	ADJ
cana-1747	6	13	variables	variable	NOUN
cana-1747	6	14	with	with	ADP
cana-1747	6	15	increasing	increase	VERB
cana-1747	6	16	sample	sample	NOUN
cana-1747	6	17	size	size	NOUN
cana-1747	6	18	,	,	PUNCT
cana-1747	6	19	and	and	CCONJ
cana-1747	6	20	the	the	DET
cana-1747	6	21	clt	clt	NOUN
cana-1747	6	22	outlines	outline	VERB
cana-1747	6	23	the	the	DET
cana-1747	6	24	distribution	distribution	NOUN
cana-1747	6	25	of	of	ADP
cana-1747	6	26	these	these	DET
cana-1747	6	27	sums	sum	NOUN
cana-1747	6	28	,	,	PUNCT
cana-1747	6	29	the	the	DET
cana-1747	6	30	lil	lil	NOUN
cana-1747	6	31	provides	provide	VERB
cana-1747	6	32	insights	insight	NOUN
cana-1747	6	33	into	into	ADP
cana-1747	6	34	the	the	DET
cana-1747	6	35	fluctuations	fluctuation	NOUN
cana-1747	6	36	of	of	ADP
cana-1747	6	37	these	these	DET
cana-1747	6	38	sums	sum	NOUN
cana-1747	6	39	,	,	PUNCT
cana-1747	6	40	especially	especially	ADV
cana-1747	6	41	concerning	concern	VERB
cana-1747	6	42	their	their	PRON
cana-1747	6	43	bounds	bound	NOUN
cana-1747	6	44	.	.	PUNCT
cana-1747	7	1	the	the	DET
cana-1747	7	2	lil	lil	NOUN
cana-1747	7	3	emerged	emerge	VERB
cana-1747	7	4	from	from	ADP
cana-1747	7	5	khintchine	khintchine	PROPN
cana-1747	7	6	's	's	PART
cana-1747	7	7	[	[	X
cana-1747	7	8	1	1	NUM
cana-1747	7	9	]	]	PUNCT
cana-1747	7	10	investigations	investigation	NOUN
cana-1747	7	11	,	,	PUNCT
cana-1747	7	12	in	in	ADP
cana-1747	7	13	which	which	PRON
cana-1747	7	14	he	he	PRON
cana-1747	7	15	sought	seek	VERB
cana-1747	7	16	to	to	PART
cana-1747	7	17	ascertain	ascertain	VERB
cana-1747	7	18	the	the	DET
cana-1747	7	19	precise	precise	ADJ
cana-1747	7	20	rate	rate	NOUN
cana-1747	7	21	of	of	ADP
cana-1747	7	22	convergence	convergence	NOUN
cana-1747	7	23	of	of	ADP
cana-1747	7	24	normal	normal	ADJ
cana-1747	7	25	numbers	number	NOUN
cana-1747	7	26	.	.	PUNCT
cana-1747	8	1	kolmogorov	kolmogorov	PROPN
cana-1747	9	1	[	[	X
cana-1747	9	2	5	5	NUM
cana-1747	9	3	]	]	PUNCT
cana-1747	9	4	later	later	ADV
cana-1747	9	5	generalized	generalize	VERB
cana-1747	9	6	this	this	DET
cana-1747	9	7	result	result	NOUN
cana-1747	9	8	to	to	PART
cana-1747	9	9	include	include	VERB
cana-1747	9	10	independent	independent	ADJ
cana-1747	9	11	random	random	ADJ
cana-1747	9	12	variables	variable	NOUN
cana-1747	9	13	.	.	PUNCT
cana-1747	10	1	since	since	SCONJ
cana-1747	10	2	its	its	PRON
cana-1747	10	3	inception	inception	NOUN
cana-1747	10	4	,	,	PUNCT
cana-1747	10	5	the	the	DET
cana-1747	10	6	lil	lil	NOUN
cana-1747	10	7	has	have	AUX
cana-1747	10	8	developed	develop	VERB
cana-1747	10	9	into	into	ADP
cana-1747	10	10	a	a	DET
cana-1747	10	11	fundamental	fundamental	ADJ
cana-1747	10	12	theorem	theorem	NOUN
cana-1747	10	13	with	with	ADP
cana-1747	10	14	extensive	extensive	ADJ
cana-1747	10	15	applications	application	NOUN
cana-1747	10	16	spanning	span	VERB
cana-1747	10	17	various	various	ADJ
cana-1747	10	18	areas	area	NOUN
cana-1747	10	19	of	of	ADP
cana-1747	10	20	mathematics	mathematic	NOUN
cana-1747	10	21	and	and	CCONJ
cana-1747	10	22	statistics	statistic	NOUN
cana-1747	10	23	.	.	PUNCT
cana-1747	11	1	a	a	DET
cana-1747	11	2	similar	similar	ADJ
cana-1747	11	3	lil	lil	NOUN
cana-1747	11	4	has	have	AUX
cana-1747	11	5	been	be	AUX
cana-1747	11	6	developed	develop	VERB
cana-1747	11	7	across	across	ADP
cana-1747	11	8	different	different	ADJ
cana-1747	11	9	fields	field	NOUN
cana-1747	11	10	,	,	PUNCT
cana-1747	11	11	including	include	VERB
cana-1747	11	12	harmonic	harmonic	ADJ
cana-1747	11	13	functions	function	NOUN
cana-1747	11	14	[	[	X
cana-1747	11	15	7	7	NUM
cana-1747	11	16	]	]	PUNCT
cana-1747	11	17	,	,	PUNCT
cana-1747	11	18	martingales	martingale	NOUN
cana-1747	11	19	[	[	X
cana-1747	11	20	10	10	NUM
cana-1747	11	21	]	]	PUNCT
cana-1747	11	22	,	,	PUNCT
cana-1747	11	23	[	[	X
cana-1747	11	24	12	12	NUM
cana-1747	11	25	]	]	PUNCT
cana-1747	11	26	,	,	PUNCT
cana-1747	11	27	q	q	ADJ
cana-1747	11	28	-	-	ADJ
cana-1747	11	29	lacunary	lacunary	ADJ
cana-1747	11	30	series	series	NOUN
cana-1747	11	31	[	[	X
cana-1747	11	32	8	8	NUM
cana-1747	11	33	]	]	PUNCT
cana-1747	11	34	,	,	PUNCT
cana-1747	11	35	random	random	ADJ
cana-1747	11	36	walks	walk	NOUN
cana-1747	11	37	,	,	PUNCT
cana-1747	11	38	stochastic	stochastic	NOUN
cana-1747	11	39	processes	process	NOUN
cana-1747	11	40	,	,	PUNCT
cana-1747	11	41	and	and	CCONJ
cana-1747	11	42	more	more	ADJ
cana-1747	11	43	.	.	PUNCT
cana-1747	12	1	in	in	ADP
cana-1747	12	2	the	the	DET
cana-1747	12	3	realm	realm	NOUN
cana-1747	12	4	of	of	ADP
cana-1747	12	5	mathematics	mathematic	NOUN
cana-1747	12	6	,	,	PUNCT
cana-1747	12	7	salem	salem	NOUN
cana-1747	12	8	and	and	CCONJ
cana-1747	12	9	zygmund	zygmund	NOUN
cana-1747	12	10	[	[	X
cana-1747	12	11	8	8	NUM
cana-1747	12	12	]	]	PUNCT
cana-1747	12	13	were	be	AUX
cana-1747	12	14	the	the	DET
cana-1747	12	15	first	first	ADJ
cana-1747	12	16	to	to	PART
cana-1747	12	17	introduce	introduce	VERB
cana-1747	12	18	a	a	DET
cana-1747	12	19	lil	lil	NOUN
cana-1747	12	20	for	for	ADP
cana-1747	12	21	the	the	DET
cana-1747	12	22	sums	sum	NOUN
cana-1747	12	23	of	of	ADP
cana-1747	12	24	q	q	ADJ
cana-1747	12	25	-	-	ADJ
cana-1747	12	26	lacunary	lacunary	ADJ
cana-1747	12	27	trigonometric	trigonometric	ADJ
cana-1747	12	28	series	series	NOUN
cana-1747	12	29	.	.	PUNCT
cana-1747	13	1	erdos	erdo	NOUN
cana-1747	13	2	and	and	CCONJ
cana-1747	13	3	gal	gal	ADJ
cana-1747	14	1	[	[	X
cana-1747	14	2	6	6	NUM
cana-1747	14	3	]	]	PUNCT
cana-1747	14	4	subsequently	subsequently	ADV
cana-1747	14	5	obtained	obtain	VERB
cana-1747	14	6	a	a	DET
cana-1747	14	7	comparable	comparable	ADJ
cana-1747	14	8	outcome	outcome	NOUN
cana-1747	14	9	for	for	ADP
cana-1747	14	10	a	a	DET
cana-1747	14	11	particular	particular	ADJ
cana-1747	14	12	category	category	NOUN
cana-1747	14	13	of	of	ADP
cana-1747	14	14	q	q	ADJ
cana-1747	14	15	-	-	ADJ
cana-1747	14	16	lacunary	lacunary	ADJ
cana-1747	14	17	series	series	NOUN
cana-1747	14	18	.	.	PUNCT
cana-1747	15	1	in	in	ADP
cana-1747	15	2	this	this	DET
cana-1747	15	3	lil	lil	NOUN
cana-1747	15	4	,	,	PUNCT
cana-1747	15	5	only	only	ADV
cana-1747	15	6	the	the	DET
cana-1747	15	7	sum	sum	NOUN
cana-1747	15	8	of	of	ADP
cana-1747	15	9	the	the	DET
cana-1747	15	10	first	first	ADJ
cana-1747	15	11	n	n	NOUN
cana-1747	15	12	-	-	PUNCT
cana-1747	15	13	terms	term	NOUN
cana-1747	15	14	of	of	ADP
cana-1747	15	15	the	the	DET
cana-1747	15	16	lacunary	lacunary	ADJ
cana-1747	15	17	series	series	NOUN
cana-1747	15	18	was	be	AUX
cana-1747	15	19	considered	consider	VERB
cana-1747	15	20	as	as	ADP
cana-1747	15	21	in	in	ADP
cana-1747	15	22	kolmogorov	kolmogorov	PROPN
cana-1747	15	23	's	's	PART
cana-1747	15	24	lil	lil	PROPN
cana-1747	15	25	.	.	PUNCT
cana-1747	16	1	subsequently	subsequently	ADV
cana-1747	16	2	,	,	PUNCT
cana-1747	16	3	m.	m.	NOUN
cana-1747	16	4	weiss	weiss	PROPN
cana-1747	17	1	[	[	X
cana-1747	17	2	4	4	NUM
cana-1747	17	3	]	]	PUNCT
cana-1747	17	4	obtained	obtain	VERB
cana-1747	17	5	an	an	DET
cana-1747	17	6	lil	lil	NOUN
cana-1747	17	7	for	for	ADP
cana-1747	17	8	q	q	ADJ
cana-1747	17	9	-	-	ADJ
cana-1747	17	10	lacunary	lacunary	ADJ
cana-1747	17	11	series	series	NOUN
cana-1747	17	12	analogous	analogous	ADJ
cana-1747	17	13	to	to	ADP
cana-1747	17	14	kolmogorov	kolmogorov	PROPN
cana-1747	17	15	's	's	PART
cana-1747	17	16	lil	lil	PROPN
cana-1747	17	17	.	.	PUNCT
cana-1747	18	1	additionally	additionally	ADV
cana-1747	18	2	,	,	PUNCT
cana-1747	18	3	in	in	ADP
cana-1747	18	4	the	the	DET
cana-1747	18	5	same	same	ADJ
cana-1747	18	6	paper	paper	NOUN
cana-1747	18	7	,	,	PUNCT
cana-1747	18	8	salem	salem	NOUN
cana-1747	18	9	and	and	CCONJ
cana-1747	18	10	zygmund	zygmund	NOUN
cana-1747	18	11	[	[	X
cana-1747	18	12	8	8	NUM
cana-1747	18	13	]	]	PUNCT
cana-1747	18	14	introduced	introduce	VERB
cana-1747	18	15	another	another	DET
cana-1747	18	16	lil	lil	NOUN
cana-1747	18	17	for	for	ADP
cana-1747	18	18	q	q	ADJ
cana-1747	18	19	-	-	ADJ
cana-1747	18	20	lacunary	lacunary	ADJ
cana-1747	18	21	series	series	NOUN
cana-1747	18	22	,	,	PUNCT
cana-1747	18	23	as	as	SCONJ
cana-1747	18	24	stated	state	VERB
cana-1747	18	25	below	below	ADV
cana-1747	18	26	:	:	PUNCT
cana-1747	18	27	theorem	theorem	NOUN
cana-1747	18	28	1	1	NUM
cana-1747	18	29	(	(	PUNCT
cana-1747	18	30	salem	salem	NOUN
cana-1747	18	31	and	and	CCONJ
cana-1747	18	32	zygmund	zygmund	NOUN
cana-1747	18	33	)	)	PUNCT
cana-1747	18	34	let	let	VERB
cana-1747	18	35	�	�	PROPN
cana-1747	18	36	̃	̃	NOUN
cana-1747	18	37	�	�	NOUN
cana-1747	18	38	𝑁	𝑁	NOUN
cana-1747	18	39	=	=	SYM
cana-1747	18	40	∑	∑	PUNCT
cana-1747	18	41	(	(	PUNCT
cana-1747	18	42	𝑎𝑖	𝑎𝑖	X
cana-1747	18	43	cos	cos	PROPN
cana-1747	18	44	𝑛𝑖𝜃	𝑛𝑖𝜃	PROPN
cana-1747	18	45	+	+	CCONJ
cana-1747	18	46	𝑏𝑖	𝑏𝑖	ADP
cana-1747	18	47	sin	sin	NOUN
cana-1747	18	48	𝑛𝑖𝜃	𝑛𝑖𝜃	PROPN
cana-1747	18	49	)	)	PUNCT
cana-1747	18	50	∞	∞	NUM
cana-1747	18	51	𝑖=𝑁	𝑖=𝑁	PUNCT
cana-1747	18	52	with	with	ADP
cana-1747	18	53	𝑛𝑖+1	𝑛𝑖+1	PROPN
cana-1747	18	54	𝑛𝑖	𝑛𝑖	PROPN
cana-1747	18	55	>	>	X
cana-1747	18	56	𝑞	𝑞	X
cana-1747	18	57	>	>	X
cana-1747	18	58	1	1	NUM
cana-1747	18	59	and	and	CCONJ
cana-1747	18	60	𝑐𝑖	𝑐𝑖	NOUN
cana-1747	18	61	2	2	NUM
cana-1747	18	62	=	=	SYM
cana-1747	18	63	𝑎𝑖	𝑎𝑖	ADP
cana-1747	18	64	2	2	NUM
cana-1747	18	65	+	+	CCONJ
cana-1747	18	66	𝑏𝑖	𝑏𝑖	PROPN
cana-1747	18	67	2	2	NUM
cana-1747	18	68	satisfy	satisfy	NOUN
cana-1747	18	69	∑	∑	ADV
cana-1747	18	70	𝑐𝑖	𝑐𝑖	NOUN
cana-1747	18	71	2∞	2∞	NUM
cana-1747	18	72	𝑖=1	𝑖=1	PUNCT
cana-1747	18	73	<	<	X
cana-1747	18	74	∞.	∞.	PROPN
cana-1747	18	75	define	define	VERB
cana-1747	18	76	�	�	PROPN
cana-1747	18	77	̃	̃	PROPN
cana-1747	18	78	�	�	PROPN
cana-1747	18	79	𝑀	𝑀	NOUN
cana-1747	18	80	=	=	PUNCT
cana-1747	18	81	∑	∑	PUNCT
cana-1747	18	82	𝑐𝑖	𝑐𝑖	PROPN
cana-1747	18	83	2∞	2∞	NUM
cana-1747	18	84	𝑖=𝑀	𝑖=𝑀	PUNCT
cana-1747	18	85	and	and	CCONJ
cana-1747	18	86	�	�	PROPN
cana-1747	18	87	̃	̃	PROPN
cana-1747	18	88	�	�	PROPN
cana-1747	18	89	𝑀	𝑀	NOUN
cana-1747	18	90	=	=	PUNCT
cana-1747	18	91	max	max	PROPN
cana-1747	18	92	i≥m	i≥m	PROPN
cana-1747	18	93	|𝑐𝑖|	|𝑐𝑖|	PROPN
cana-1747	18	94	.	.	PROPN
cana-1747	19	1	assume	assume	VERB
cana-1747	19	2	that	that	SCONJ
cana-1747	19	3	�	�	PROPN
cana-1747	19	4	̃	̃	NOUN
cana-1747	19	5	�	�	NOUN
cana-1747	19	6	1	1	NUM
cana-1747	19	7	<	<	X
cana-1747	19	8	∞	∞	PROPN
cana-1747	19	9	and	and	CCONJ
cana-1747	19	10	�	�	PROPN
cana-1747	19	11	̃	̃	PROPN
cana-1747	19	12	�	�	PROPN
cana-1747	19	13	𝑀	𝑀	PROPN
cana-1747	19	14	2	2	NUM
cana-1747	19	15	≤	≤	NOUN
cana-1747	19	16	𝐾𝑀	𝐾𝑀	PROPN
cana-1747	19	17	(	(	PUNCT
cana-1747	19	18	�	�	NOUN
cana-1747	19	19	̃	̃	PROPN
cana-1747	19	20	�	�	NOUN
cana-1747	19	21	𝑀	𝑀	NOUN
cana-1747	19	22	2	2	NUM
cana-1747	19	23	ln	ln	NOUN
cana-1747	19	24	ln	ln	ADJ
cana-1747	19	25	1	1	NUM
cana-1747	19	26	�	�	PROPN
cana-1747	19	27	̃	̃	PROPN
cana-1747	19	28	�	�	PROPN
cana-1747	19	29	𝑀	𝑀	PROPN
cana-1747	19	30	)	)	PUNCT
cana-1747	19	31	where	where	SCONJ
cana-1747	19	32	𝐾𝑀	𝐾𝑀	PROPN
cana-1747	19	33	approaches	approach	VERB
cana-1747	19	34	to	to	ADP
cana-1747	19	35	0	0	NUM
cana-1747	19	36	and	and	CCONJ
cana-1747	19	37	m	m	PROPN
cana-1747	19	38	approaches	approach	NOUN
cana-1747	19	39	to	to	ADP
cana-1747	19	40	infinity	infinity	NOUN
cana-1747	19	41	.	.	PUNCT
cana-1747	20	1	then	then	ADV
cana-1747	20	2	communications	communication	NOUN
cana-1747	20	3	on	on	ADP
cana-1747	20	4	applied	apply	VERB
cana-1747	20	5	nonlinear	nonlinear	ADJ
cana-1747	20	6	analysis	analysis	NOUN
cana-1747	20	7	issn	issn	NOUN
cana-1747	20	8	:	:	PUNCT
cana-1747	20	9	1074	1074	NUM
cana-1747	20	10	-	-	PUNCT
cana-1747	20	11	133x	133x	NUM
cana-1747	20	12	vol	vol	NOUN
cana-1747	20	13	32	32	NUM
cana-1747	20	14	no	no	NOUN
cana-1747	20	15	.	.	NOUN
cana-1747	20	16	2	2	NUM
cana-1747	20	17	(	(	PUNCT
cana-1747	20	18	2025	2025	NUM
cana-1747	20	19	)	)	PUNCT
cana-1747	20	20	356	356	NUM
cana-1747	20	21	https://internationalpubls.com	https://internationalpubls.com	X
cana-1747	20	22	limsup	limsup	PROPN
cana-1747	20	23	m→∞	m→∞	NOUN
cana-1747	20	24	�	�	PROPN
cana-1747	20	25	̃	̃	PROPN
cana-1747	20	26	�	�	NOUN
cana-1747	20	27	𝑀(𝜃	𝑀(𝜃	NUM
cana-1747	20	28	)	)	PUNCT
cana-1747	20	29	√2	√2	PROPN
cana-1747	20	30	�	�	PROPN
cana-1747	20	31	̃	̃	PROPN
cana-1747	20	32	�	�	PROPN
cana-1747	20	33	𝑀	𝑀	PROPN
cana-1747	20	34	2	2	NUM
cana-1747	20	35	ln	ln	NOUN
cana-1747	20	36	ln	ln	ADJ
cana-1747	20	37	1	1	NUM
cana-1747	20	38	�	�	PROPN
cana-1747	20	39	̃	̃	PROPN
cana-1747	20	40	�	�	PROPN
cana-1747	20	41	𝑀	𝑀	NOUN
cana-1747	20	42	≤	≤	NOUN
cana-1747	20	43	1	1	NUM
cana-1747	20	44	for	for	ADP
cana-1747	20	45	a.e	a.e	PROPN
cana-1747	20	46	.	.	PROPN
cana-1747	20	47	𝜃	𝜃	PROPN
cana-1747	20	48	in	in	ADP
cana-1747	20	49	the	the	DET
cana-1747	20	50	unit	unit	NOUN
cana-1747	20	51	circle	circle	NOUN
cana-1747	20	52	.	.	PUNCT
cana-1747	21	1	in	in	ADP
cana-1747	21	2	this	this	DET
cana-1747	21	3	lil	lil	NOUN
cana-1747	21	4	variant	variant	NOUN
cana-1747	21	5	,	,	PUNCT
cana-1747	21	6	the	the	DET
cana-1747	21	7	focus	focus	NOUN
cana-1747	21	8	lies	lie	VERB
cana-1747	21	9	on	on	ADP
cana-1747	21	10	the	the	DET
cana-1747	21	11	sums	sum	NOUN
cana-1747	21	12	beyond	beyond	ADP
cana-1747	21	13	the	the	DET
cana-1747	21	14	initial	initial	ADJ
cana-1747	21	15	n	n	NOUN
cana-1747	21	16	-	-	PUNCT
cana-1747	21	17	terms	term	NOUN
cana-1747	21	18	,	,	PUNCT
cana-1747	21	19	specifically	specifically	ADV
cana-1747	21	20	on	on	ADP
cana-1747	21	21	the	the	DET
cana-1747	21	22	tail	tail	NOUN
cana-1747	21	23	sums	sum	NOUN
cana-1747	21	24	of	of	ADP
cana-1747	21	25	the	the	DET
cana-1747	21	26	series	series	NOUN
cana-1747	21	27	.	.	PUNCT
cana-1747	22	1	consequently	consequently	ADV
cana-1747	22	2	,	,	PUNCT
cana-1747	22	3	this	this	DET
cana-1747	22	4	version	version	NOUN
cana-1747	22	5	is	be	AUX
cana-1747	22	6	often	often	ADV
cana-1747	22	7	referred	refer	VERB
cana-1747	22	8	to	to	ADP
cana-1747	22	9	as	as	ADP
cana-1747	22	10	the	the	DET
cana-1747	22	11	`	`	PUNCT
cana-1747	22	12	`	`	PUNCT
cana-1747	22	13	tail	tail	NOUN
cana-1747	22	14	lil	lil	NOUN
cana-1747	22	15	"	"	PUNCT
cana-1747	22	16	because	because	SCONJ
cana-1747	22	17	of	of	ADP
cana-1747	22	18	its	its	PRON
cana-1747	22	19	emphasis	emphasis	NOUN
cana-1747	22	20	on	on	ADP
cana-1747	22	21	the	the	DET
cana-1747	22	22	tail	tail	NOUN
cana-1747	22	23	sum	sum	NOUN
cana-1747	22	24	aspect	aspect	NOUN
cana-1747	22	25	.	.	PUNCT
cana-1747	23	1	salem	salem	NOUN
cana-1747	23	2	and	and	CCONJ
cana-1747	23	3	zygmund	zygmund	NOUN
cana-1747	23	4	only	only	ADV
cana-1747	23	5	derived	derive	VERB
cana-1747	23	6	the	the	DET
cana-1747	23	7	upper	upper	ADJ
cana-1747	23	8	bound	bind	VERB
cana-1747	23	9	in	in	ADP
cana-1747	23	10	this	this	DET
cana-1747	23	11	tail	tail	NOUN
cana-1747	23	12	lil	lil	NOUN
cana-1747	23	13	.	.	PUNCT
cana-1747	24	1	under	under	ADP
cana-1747	24	2	similar	similar	ADJ
cana-1747	24	3	conditions	condition	NOUN
cana-1747	24	4	,	,	PUNCT
cana-1747	24	5	s.	s.	PROPN
cana-1747	24	6	ghimire	ghimire	PROPN
cana-1747	24	7	and	and	CCONJ
cana-1747	24	8	c.n	c.n	PROPN
cana-1747	24	9	.	.	PROPN
cana-1747	24	10	moore	moore	PROPN
cana-1747	25	1	[	[	X
cana-1747	25	2	9	9	NUM
cana-1747	25	3	]	]	PUNCT
cana-1747	25	4	established	establish	VERB
cana-1747	25	5	the	the	DET
cana-1747	25	6	converse	converse	NOUN
cana-1747	25	7	of	of	ADP
cana-1747	25	8	the	the	DET
cana-1747	25	9	aforementioned	aforementioned	ADJ
cana-1747	25	10	result	result	NOUN
cana-1747	25	11	.	.	PUNCT
cana-1747	26	1	their	their	PRON
cana-1747	26	2	result	result	NOUN
cana-1747	26	3	is	be	AUX
cana-1747	26	4	:	:	PUNCT
cana-1747	26	5	theorem	theorem	ADJ
cana-1747	26	6	2	2	NUM
cana-1747	26	7	.	.	PUNCT
cana-1747	26	8	assuming	assume	VERB
cana-1747	26	9	the	the	DET
cana-1747	26	10	same	same	ADJ
cana-1747	26	11	notation	notation	NOUN
cana-1747	26	12	and	and	CCONJ
cana-1747	26	13	hypotheses	hypothesis	NOUN
cana-1747	26	14	as	as	SCONJ
cana-1747	26	15	stated	state	VERB
cana-1747	26	16	in	in	ADP
cana-1747	26	17	the	the	DET
cana-1747	26	18	preceding	precede	VERB
cana-1747	26	19	theorem	theorem	NOUN
cana-1747	26	20	,	,	PUNCT
cana-1747	26	21	we	we	PRON
cana-1747	26	22	have	have	VERB
cana-1747	26	23	limsup	limsup	ADJ
cana-1747	26	24	n→∞	n→∞	X
cana-1747	26	25	�	�	PROPN
cana-1747	26	26	̃	̃	PROPN
cana-1747	26	27	�	�	NOUN
cana-1747	26	28	𝑁(𝜃	𝑁(𝜃	NOUN
cana-1747	26	29	)	)	PUNCT
cana-1747	26	30	√2	√2	PROPN
cana-1747	26	31	�	�	PROPN
cana-1747	26	32	̃	̃	NOUN
cana-1747	26	33	�	�	NOUN
cana-1747	26	34	𝑁	𝑁	ADJ
cana-1747	26	35	2	2	NUM
cana-1747	26	36	ln	ln	NOUN
cana-1747	26	37	ln	ln	ADJ
cana-1747	26	38	1	1	NUM
cana-1747	26	39	�	�	PROPN
cana-1747	26	40	̃	̃	NOUN
cana-1747	26	41	�	�	NOUN
cana-1747	26	42	𝑁	𝑁	NOUN
cana-1747	26	43	≥	≥	NOUN
cana-1747	26	44	1	1	NUM
cana-1747	26	45	for	for	ADP
cana-1747	26	46	a.e	a.e	PROPN
cana-1747	26	47	.	.	PROPN
cana-1747	26	48	𝜃	𝜃	NOUN
cana-1747	26	49	in	in	ADP
cana-1747	26	50	[	[	X
cana-1747	26	51	0	0	NUM
cana-1747	26	52	,	,	PUNCT
cana-1747	26	53	2𝜋	2𝜋	NUM
cana-1747	26	54	]	]	PUNCT
cana-1747	26	55	.	.	PUNCT
cana-1747	27	1	now	now	ADV
cana-1747	27	2	theorem	theorem	VERB
cana-1747	27	3	2	2	NUM
cana-1747	27	4	,	,	PUNCT
cana-1747	27	5	when	when	SCONJ
cana-1747	27	6	combined	combine	VERB
cana-1747	27	7	with	with	ADP
cana-1747	27	8	theorem	theorem	ADJ
cana-1747	27	9	1	1	NUM
cana-1747	27	10	give	give	VERB
cana-1747	27	11	the	the	DET
cana-1747	27	12	conclusion	conclusion	NOUN
cana-1747	27	13	limsup	limsup	X
cana-1747	27	14	n→∞	n→∞	X
cana-1747	27	15	�	�	PROPN
cana-1747	27	16	̃	̃	PROPN
cana-1747	27	17	�	�	NOUN
cana-1747	27	18	𝑁(𝜃	𝑁(𝜃	NOUN
cana-1747	27	19	)	)	PUNCT
cana-1747	27	20	√2	√2	PROPN
cana-1747	27	21	�	�	PROPN
cana-1747	27	22	̃	̃	NOUN
cana-1747	27	23	�	�	NOUN
cana-1747	27	24	𝑁	𝑁	ADJ
cana-1747	27	25	2	2	NUM
cana-1747	27	26	ln	ln	NOUN
cana-1747	27	27	ln	ln	ADJ
cana-1747	27	28	1	1	NUM
cana-1747	27	29	�	�	PROPN
cana-1747	27	30	̃	̃	NOUN
cana-1747	27	31	�	�	NOUN
cana-1747	27	32	𝑁	𝑁	NOUN
cana-1747	27	33	=	=	SYM
cana-1747	27	34	1	1	NUM
cana-1747	27	35	for	for	ADP
cana-1747	27	36	a.e	a.e	PROPN
cana-1747	27	37	.	.	PROPN
cana-1747	27	38	𝜃	𝜃	NOUN
cana-1747	27	39	in	in	ADP
cana-1747	27	40	[	[	X
cana-1747	27	41	0	0	NUM
cana-1747	27	42	,	,	PUNCT
cana-1747	27	43	2𝜋	2𝜋	NUM
cana-1747	27	44	]	]	PUNCT
cana-1747	27	45	.	.	PUNCT
cana-1747	28	1	a	a	DET
cana-1747	28	2	similar	similar	ADJ
cana-1747	28	3	lil	lil	NOUN
cana-1747	28	4	for	for	ADP
cana-1747	28	5	summation	summation	NOUN
cana-1747	28	6	of	of	ADP
cana-1747	28	7	signum	signum	PROPN
cana-1747	28	8	functions	function	NOUN
cana-1747	28	9	was	be	AUX
cana-1747	28	10	recently	recently	ADV
cana-1747	28	11	obtained	obtain	VERB
cana-1747	28	12	by	by	ADP
cana-1747	28	13	s.	s.	PROPN
cana-1747	28	14	ghimire	ghimire	PROPN
cana-1747	29	1	[	[	X
cana-1747	29	2	11	11	NUM
cana-1747	29	3	]	]	PUNCT
cana-1747	29	4	showing	show	VERB
cana-1747	29	5	that	that	SCONJ
cana-1747	29	6	the	the	DET
cana-1747	29	7	convergence	convergence	NOUN
cana-1747	29	8	rate	rate	NOUN
cana-1747	29	9	of	of	ADP
cana-1747	29	10	summation	summation	NOUN
cana-1747	29	11	of	of	ADP
cana-1747	29	12	the	the	DET
cana-1747	29	13	functions	function	NOUN
cana-1747	29	14	is	be	AUX
cana-1747	29	15	controlled	control	VERB
cana-1747	29	16	by	by	ADP
cana-1747	29	17	the	the	DET
cana-1747	29	18	tail	tail	NOUN
cana-1747	29	19	sums	sum	NOUN
cana-1747	29	20	of	of	ADP
cana-1747	29	21	the	the	DET
cana-1747	29	22	square	square	ADJ
cana-1747	29	23	function	function	NOUN
cana-1747	29	24	as	as	ADP
cana-1747	29	25	in	in	ADP
cana-1747	29	26	the	the	DET
cana-1747	29	27	lil	lil	NOUN
cana-1747	29	28	introduced	introduce	VERB
cana-1747	29	29	by	by	ADP
cana-1747	29	30	salem	salem	NOUN
cana-1747	29	31	and	and	CCONJ
cana-1747	29	32	zygmund	zygmund	NOUN
cana-1747	29	33	.	.	PUNCT
cana-1747	30	1	the	the	DET
cana-1747	30	2	one	one	NUM
cana-1747	30	3	-	-	PUNCT
cana-1747	30	4	sided	sided	ADJ
cana-1747	30	5	version	version	NOUN
cana-1747	30	6	of	of	ADP
cana-1747	30	7	s.	s.	PROPN
cana-1747	30	8	ghimire	ghimire	PROPN
cana-1747	30	9	's	's	PART
cana-1747	30	10	lil	lil	NOUN
cana-1747	30	11	is	be	AUX
cana-1747	30	12	as	as	SCONJ
cana-1747	30	13	follows	follow	VERB
cana-1747	30	14	:	:	PUNCT
cana-1747	30	15	theorem	theorem	NOUN
cana-1747	30	16	3	3	X
cana-1747	30	17	.	.	PUNCT
cana-1747	30	18	suppose	suppose	VERB
cana-1747	30	19	{	{	PUNCT
cana-1747	30	20	𝑢𝑖	𝑢𝑖	NOUN
cana-1747	30	21	}	}	PUNCT
cana-1747	30	22	is	be	AUX
cana-1747	30	23	a	a	DET
cana-1747	30	24	sequence	sequence	NOUN
cana-1747	30	25	of	of	ADP
cana-1747	30	26	signum	signum	ADJ
cana-1747	30	27	functions	function	NOUN
cana-1747	30	28	defined	define	VERB
cana-1747	30	29	by	by	ADP
cana-1747	30	30	𝑢𝑖(𝑡	𝑢𝑖(𝑡	NUM
cana-1747	30	31	)	)	PUNCT
cana-1747	30	32	=	=	VERB
cana-1747	31	1	𝑠𝑔𝑛	𝑠𝑔𝑛	PROPN
cana-1747	31	2	(	(	PUNCT
cana-1747	31	3	sin	sin	NOUN
cana-1747	31	4	2	2	NUM
cana-1747	31	5	𝑖𝜋𝑡	𝑖𝜋𝑡	NOUN
cana-1747	31	6	)	)	PUNCT
cana-1747	31	7	and	and	CCONJ
cana-1747	31	8	{	{	PUNCT
cana-1747	31	9	𝑏𝑖}𝑖=1	𝑏𝑖}𝑖=1	NOUN
cana-1747	31	10	∞	∞	PROPN
cana-1747	31	11	where	where	SCONJ
cana-1747	31	12	𝑏𝑖	𝑏𝑖	PROPN
cana-1747	31	13	∈	∈	PROPN
cana-1747	31	14	ℝ	ℝ	PROPN
cana-1747	31	15	satisfies	satisfy	VERB
cana-1747	31	16	∑	∑	ADV
cana-1747	31	17	𝑏𝑖	𝑏𝑖	ADP
cana-1747	31	18	2	2	NUM
cana-1747	31	19	<	<	X
cana-1747	31	20	∞.∞	∞.∞	PROPN
cana-1747	31	21	𝑖=1	𝑖=1	PROPN
cana-1747	31	22	then	then	ADV
cana-1747	31	23	limsup	limsup	X
cana-1747	31	24	n→∞	n→∞	X
cana-1747	32	1	|	|	ADV
cana-1747	32	2	∑	∑	PUNCT
cana-1747	32	3	𝑏𝑖𝑢𝑖(𝑡)|	𝑏𝑖𝑢𝑖(𝑡)|	NOUN
cana-1747	32	4	∞	∞	NUM
cana-1747	32	5	𝑖=𝑛+1	𝑖=𝑛+1	NOUN
cana-1747	32	6	√2∑	√2∑	NOUN
cana-1747	32	7	𝑏𝑖	𝑏𝑖	ADP
cana-1747	32	8	2	2	NUM
cana-1747	32	9	∞	∞	NUM
cana-1747	32	10	𝑖=𝑛+1	𝑖=𝑛+1	NOUN
cana-1747	32	11	ln	ln	NOUN
cana-1747	32	12	ln	ln	ADJ
cana-1747	33	1	(	(	PUNCT
cana-1747	33	2	1	1	NUM
cana-1747	33	3	∑	∑	ADV
cana-1747	33	4	𝑏𝑖	𝑏𝑖	ADP
cana-1747	33	5	2	2	NUM
cana-1747	33	6	∞	∞	NUM
cana-1747	33	7	𝑖=𝑛+1	𝑖=𝑛+1	NOUN
cana-1747	33	8	)	)	PUNCT
cana-1747	33	9	≤	≤	NUM
cana-1747	33	10	1	1	NUM
cana-1747	33	11	for	for	ADP
cana-1747	33	12	a.e	a.e	PROPN
cana-1747	33	13	.	.	PUNCT
cana-1747	33	14	𝑡	𝑡	PROPN
cana-1747	33	15	∈	∈	PROPN
cana-1747	34	1	[	[	X
cana-1747	34	2	0	0	NUM
cana-1747	34	3	,	,	PUNCT
cana-1747	34	4	1	1	NUM
cana-1747	34	5	)	)	PUNCT
cana-1747	34	6	.	.	PUNCT
cana-1747	35	1	here	here	ADV
cana-1747	35	2	,	,	PUNCT
cana-1747	35	3	we	we	PRON
cana-1747	35	4	obtain	obtain	VERB
cana-1747	35	5	the	the	DET
cana-1747	35	6	lower	low	ADJ
cana-1747	35	7	limit	limit	NOUN
cana-1747	35	8	version	version	NOUN
cana-1747	35	9	of	of	ADP
cana-1747	35	10	the	the	DET
cana-1747	35	11	lil	lil	NOUN
cana-1747	35	12	for	for	ADP
cana-1747	35	13	the	the	DET
cana-1747	35	14	summation	summation	NOUN
cana-1747	35	15	of	of	ADP
cana-1747	35	16	signum	signum	PROPN
cana-1747	35	17	functions	function	NOUN
cana-1747	35	18	.	.	PUNCT
cana-1747	36	1	our	our	PRON
cana-1747	36	2	main	main	ADJ
cana-1747	36	3	result	result	NOUN
cana-1747	36	4	is	be	AUX
cana-1747	36	5	:	:	PUNCT
cana-1747	36	6	theorem	theorem	ADJ
cana-1747	36	7	4	4	NUM
cana-1747	36	8	.	.	PUNCT
cana-1747	36	9	suppose	suppose	VERB
cana-1747	36	10	{	{	PUNCT
cana-1747	36	11	𝑢𝑖	𝑢𝑖	NOUN
cana-1747	36	12	}	}	PUNCT
cana-1747	36	13	is	be	AUX
cana-1747	36	14	a	a	DET
cana-1747	36	15	sequence	sequence	NOUN
cana-1747	36	16	of	of	ADP
cana-1747	36	17	signum	signum	ADJ
cana-1747	36	18	functions	function	NOUN
cana-1747	36	19	defined	define	VERB
cana-1747	36	20	by	by	ADP
cana-1747	36	21	𝑢𝑖(𝑡	𝑢𝑖(𝑡	NUM
cana-1747	36	22	)	)	PUNCT
cana-1747	36	23	=	=	VERB
cana-1747	36	24	𝑠𝑔𝑛	𝑠𝑔𝑛	PROPN
cana-1747	36	25	(	(	PUNCT
cana-1747	36	26	sin	sin	NOUN
cana-1747	36	27	2	2	NUM
cana-1747	36	28	𝑖𝜋𝑡	𝑖𝜋𝑡	NOUN
cana-1747	36	29	)	)	PUNCT
cana-1747	36	30	and	and	CCONJ
cana-1747	36	31	{	{	PUNCT
cana-1747	36	32	𝑏𝑖}𝑖=1	𝑏𝑖}𝑖=1	NOUN
cana-1747	36	33	∞	∞	PROPN
cana-1747	36	34	where	where	SCONJ
cana-1747	36	35	{	{	PUNCT
cana-1747	36	36	𝑏𝑖	𝑏𝑖	AUX
cana-1747	36	37	}	}	PUNCT
cana-1747	36	38	is	be	AUX
cana-1747	36	39	a	a	DET
cana-1747	36	40	square	square	ADJ
cana-1747	36	41	integrable	integrable	ADJ
cana-1747	36	42	real	real	ADV
cana-1747	36	43	-	-	PUNCT
cana-1747	36	44	valued	value	VERB
cana-1747	36	45	sequence	sequence	NOUN
cana-1747	36	46	with	with	ADP
cana-1747	36	47	𝐵𝑛	𝐵𝑛	PROPN
cana-1747	36	48	=	=	PUNCT
cana-1747	36	49	∑	∑	PROPN
cana-1747	36	50	𝑏𝑖	𝑏𝑖	ADP
cana-1747	36	51	2∞	2∞	NUM
cana-1747	36	52	𝑖=𝑛	𝑖=𝑛	PROPN
cana-1747	36	53	and	and	CCONJ
cana-1747	36	54	assume	assume	VERB
cana-1747	36	55	lim	lim	PROPN
cana-1747	36	56	n→∞	n→∞	NUM
cana-1747	36	57	𝑏𝑛	𝑏𝑛	ADP
cana-1747	36	58	2	2	NUM
cana-1747	36	59	𝐵𝑛	𝐵𝑛	PROPN
cana-1747	36	60	=	=	NOUN
cana-1747	36	61	0	0	X
cana-1747	36	62	.	.	PUNCT
cana-1747	37	1	then	then	ADV
cana-1747	37	2	communications	communication	NOUN
cana-1747	37	3	on	on	ADP
cana-1747	37	4	applied	apply	VERB
cana-1747	37	5	nonlinear	nonlinear	ADJ
cana-1747	37	6	analysis	analysis	NOUN
cana-1747	37	7	issn	issn	NOUN
cana-1747	37	8	:	:	PUNCT
cana-1747	37	9	1074	1074	NUM
cana-1747	37	10	-	-	PUNCT
cana-1747	37	11	133x	133x	NUM
cana-1747	37	12	vol	vol	NOUN
cana-1747	37	13	32	32	NUM
cana-1747	37	14	no	no	NOUN
cana-1747	37	15	.	.	NOUN
cana-1747	37	16	2	2	NUM
cana-1747	37	17	(	(	PUNCT
cana-1747	37	18	2025	2025	NUM
cana-1747	37	19	)	)	PUNCT
cana-1747	37	20	357	357	NUM
cana-1747	37	21	https://internationalpubls.com	https://internationalpubls.com	X
cana-1747	37	22	limsup	limsup	X
cana-1747	37	23	n→∞	n→∞	X
cana-1747	38	1	|	|	ADV
cana-1747	38	2	∑	∑	PUNCT
cana-1747	38	3	𝑏𝑖𝑢𝑖(𝑡)|	𝑏𝑖𝑢𝑖(𝑡)|	NOUN
cana-1747	38	4	∞	∞	NUM
cana-1747	38	5	𝑖=𝑛+1	𝑖=𝑛+1	PROPN
cana-1747	38	6	√2𝐵𝑛	√2𝐵𝑛	NOUN
cana-1747	39	1	ln	ln	PROPN
cana-1747	39	2	ln	ln	ADJ
cana-1747	39	3	1	1	NUM
cana-1747	39	4	𝐵𝑛	𝐵𝑛	PROPN
cana-1747	39	5	≥	≥	NUM
cana-1747	39	6	1	1	NUM
cana-1747	39	7	for	for	ADP
cana-1747	39	8	a.e	a.e	PROPN
cana-1747	39	9	.	.	PUNCT
cana-1747	39	10	𝑡	𝑡	PROPN
cana-1747	39	11	∈	∈	PROPN
cana-1747	40	1	[	[	X
cana-1747	40	2	0	0	NUM
cana-1747	40	3	,	,	PUNCT
cana-1747	40	4	1	1	NUM
cana-1747	40	5	)	)	PUNCT
cana-1747	40	6	.	.	PUNCT
cana-1747	41	1	note	note	VERB
cana-1747	41	2	that	that	SCONJ
cana-1747	41	3	if	if	SCONJ
cana-1747	41	4	the	the	DET
cana-1747	41	5	sequence	sequence	NOUN
cana-1747	41	6	{	{	PUNCT
cana-1747	41	7	|𝑏𝑖|	|𝑏𝑖|	PROPN
cana-1747	41	8	}	}	PUNCT
cana-1747	41	9	is	be	AUX
cana-1747	41	10	non	non	NOUN
cana-1747	41	11	increasing	increase	VERB
cana-1747	41	12	,	,	PUNCT
cana-1747	41	13	or	or	CCONJ
cana-1747	41	14	more	more	ADV
cana-1747	41	15	generally	generally	ADV
cana-1747	41	16	,	,	PUNCT
cana-1747	41	17	if	if	SCONJ
cana-1747	41	18	lim	lim	PROPN
cana-1747	41	19	i→∞	i→∞	VERB
cana-1747	41	20	|	|	ADV
cana-1747	41	21	𝑏𝑖	𝑏𝑖	INTJ
cana-1747	41	22	𝑏𝑖+1	𝑏𝑖+1	ADP
cana-1747	41	23	|	|	ADV
cana-1747	41	24	≥	≥	NOUN
cana-1747	41	25	1	1	NUM
cana-1747	41	26	,	,	PUNCT
cana-1747	41	27	then	then	ADV
cana-1747	41	28	for	for	ADP
cana-1747	41	29	large	large	ADJ
cana-1747	41	30	𝑛	𝑛	NOUN
cana-1747	41	31	,	,	PUNCT
cana-1747	41	32	we	we	PRON
cana-1747	41	33	have	have	VERB
cana-1747	41	34	𝑏𝑛	𝑏𝑛	ADP
cana-1747	41	35	2	2	NUM
cana-1747	41	36	𝐵𝑛2	𝐵𝑛2	NOUN
cana-1747	41	37	≤	≤	NOUN
cana-1747	41	38	𝑏𝑛	𝑏𝑛	ADP
cana-1747	41	39	2	2	NUM
cana-1747	41	40	∑	∑	NOUN
cana-1747	41	41	𝑏𝑖	𝑏𝑖	ADP
cana-1747	41	42	2∞	2∞	NUM
cana-1747	41	43	𝑖=𝑚	𝑖=𝑚	PROPN
cana-1747	41	44	≤	≤	ADV
cana-1747	41	45	1	1	NUM
cana-1747	41	46	𝑛	𝑛	PRON
cana-1747	41	47	+	+	NUM
cana-1747	41	48	1	1	NUM
cana-1747	41	49	and	and	CCONJ
cana-1747	41	50	assumption	assumption	NOUN
cana-1747	41	51	in	in	ADP
cana-1747	41	52	the	the	DET
cana-1747	41	53	theorem	theorem	NOUN
cana-1747	41	54	is	be	AUX
cana-1747	41	55	satisfied	satisfied	ADJ
cana-1747	41	56	.	.	PUNCT
cana-1747	42	1	the	the	DET
cana-1747	42	2	proof	proof	NOUN
cana-1747	42	3	consists	consist	VERB
cana-1747	42	4	of	of	ADP
cana-1747	42	5	stopping	stop	VERB
cana-1747	42	6	time	time	NOUN
cana-1747	42	7	argument	argument	NOUN
cana-1747	42	8	with	with	ADP
cana-1747	42	9	application	application	NOUN
cana-1747	42	10	of	of	ADP
cana-1747	42	11	certain	certain	ADJ
cana-1747	42	12	estimates	estimate	NOUN
cana-1747	42	13	.	.	PUNCT
cana-1747	43	1	to	to	PART
cana-1747	43	2	prove	prove	VERB
cana-1747	43	3	our	our	PRON
cana-1747	43	4	main	main	ADJ
cana-1747	43	5	result	result	NOUN
cana-1747	43	6	,	,	PUNCT
cana-1747	43	7	we	we	PRON
cana-1747	43	8	first	first	ADV
cana-1747	43	9	utilize	utilize	VERB
cana-1747	43	10	sub	sub	ADJ
cana-1747	43	11	-	-	ADJ
cana-1747	43	12	gaussian	gaussian	ADJ
cana-1747	43	13	type	type	NOUN
cana-1747	43	14	estimates	estimate	NOUN
cana-1747	43	15	for	for	ADP
cana-1747	43	16	the	the	DET
cana-1747	43	17	summation	summation	NOUN
cana-1747	43	18	of	of	ADP
cana-1747	43	19	signum	signum	PROPN
cana-1747	43	20	functions	function	NOUN
cana-1747	43	21	followed	follow	VERB
cana-1747	43	22	by	by	ADP
cana-1747	43	23	the	the	DET
cana-1747	43	24	application	application	NOUN
cana-1747	43	25	of	of	ADP
cana-1747	43	26	both	both	DET
cana-1747	43	27	versions	version	NOUN
cana-1747	43	28	of	of	ADP
cana-1747	43	29	borelli	borelli	PROPN
cana-1747	43	30	lemma	lemma	PROPN
cana-1747	43	31	.	.	PUNCT
cana-1747	44	1	in	in	ADP
cana-1747	44	2	what	what	PRON
cana-1747	44	3	follows	follow	VERB
cana-1747	44	4	,	,	PUNCT
cana-1747	44	5	we	we	PRON
cana-1747	44	6	use	use	VERB
cana-1747	44	7	measure	measure	NOUN
cana-1747	44	8	space	space	NOUN
cana-1747	44	9	(	(	PUNCT
cana-1747	44	10	𝐼	𝐼	NOUN
cana-1747	44	11	=	=	SYM
cana-1747	44	12	(	(	PUNCT
cana-1747	44	13	0,1	0,1	NUM
cana-1747	44	14	)	)	PUNCT
cana-1747	44	15	,	,	PUNCT
cana-1747	44	16	ℬ	ℬ	NOUN
cana-1747	44	17	,	,	PUNCT
cana-1747	44	18	𝜇	𝜇	ADP
cana-1747	44	19	)	)	PUNCT
cana-1747	44	20	and	and	CCONJ
cana-1747	44	21	|	|	ADV
cana-1747	44	22	.	.	PUNCT
cana-1747	45	1	|	|	ADV
cana-1747	45	2	stands	stand	VERB
cana-1747	45	3	for	for	ADP
cana-1747	45	4	probability	probability	NOUN
cana-1747	45	5	measure	measure	NOUN
cana-1747	45	6	𝜇	𝜇	ADP
cana-1747	45	7	restricted	restrict	VERB
cana-1747	45	8	on	on	ADP
cana-1747	45	9	i.	i.	NOUN
cana-1747	45	10	to	to	PART
cana-1747	45	11	establish	establish	VERB
cana-1747	45	12	our	our	PRON
cana-1747	45	13	main	main	ADJ
cana-1747	45	14	result	result	NOUN
cana-1747	45	15	,	,	PUNCT
cana-1747	45	16	we	we	PRON
cana-1747	45	17	begin	begin	VERB
cana-1747	45	18	by	by	ADP
cana-1747	45	19	introducing	introduce	VERB
cana-1747	45	20	some	some	DET
cana-1747	45	21	definitions	definition	NOUN
cana-1747	45	22	and	and	CCONJ
cana-1747	45	23	obtaining	obtain	VERB
cana-1747	45	24	estimates	estimate	NOUN
cana-1747	45	25	.	.	PUNCT
cana-1747	46	1	2	2	X
cana-1747	46	2	.	.	X
cana-1747	46	3	preliminaries	preliminary	NOUN
cana-1747	46	4	let	let	VERB
cana-1747	46	5	's	us	PRON
cana-1747	46	6	revisit	revisit	VERB
cana-1747	46	7	the	the	DET
cana-1747	46	8	definition	definition	NOUN
cana-1747	46	9	of	of	ADP
cana-1747	46	10	a	a	DET
cana-1747	46	11	general	general	ADJ
cana-1747	46	12	signum	signum	NOUN
cana-1747	46	13	function	function	NOUN
cana-1747	46	14	:	:	PUNCT
cana-1747	46	15	𝑠𝑔𝑛(𝑡	𝑠𝑔𝑛(𝑡	X
cana-1747	46	16	)	)	PUNCT
cana-1747	47	1	=	=	PRON
cana-1747	47	2	{	{	PUNCT
cana-1747	47	3	1	1	NUM
cana-1747	47	4	𝑖𝑓	𝑖𝑓	NUM
cana-1747	47	5	𝑡	𝑡	PROPN
cana-1747	47	6	≥	≥	NOUN
cana-1747	47	7	0	0	NUM
cana-1747	47	8	;	;	PUNCT
cana-1747	47	9	−1	−1	NOUN
cana-1747	47	10	𝑖𝑓	𝑖𝑓	NOUN
cana-1747	47	11	𝑡	𝑡	X
cana-1747	47	12	<	<	X
cana-1747	47	13	0	0	NUM
cana-1747	47	14	.	.	PUNCT
cana-1747	48	1	in	in	ADP
cana-1747	48	2	constructing	construct	VERB
cana-1747	48	3	a	a	DET
cana-1747	48	4	sequence	sequence	NOUN
cana-1747	48	5	,	,	PUNCT
cana-1747	48	6	we	we	PRON
cana-1747	48	7	define	define	VERB
cana-1747	48	8	𝑢𝑖(𝑡	𝑢𝑖(𝑡	PUNCT
cana-1747	48	9	)	)	PUNCT
cana-1747	48	10	=	=	SYM
cana-1747	48	11	𝑠𝑔𝑛(sin	𝑠𝑔𝑛(sin	VERB
cana-1747	48	12	2𝑖𝜋𝑡	2𝑖𝜋𝑡	NUM
cana-1747	48	13	)	)	PUNCT
cana-1747	48	14	on	on	ADP
cana-1747	48	15	the	the	DET
cana-1747	48	16	interval	interval	NOUN
cana-1747	48	17	(	(	PUNCT
cana-1747	48	18	0	0	NUM
cana-1747	48	19	,	,	PUNCT
cana-1747	48	20	1	1	NUM
cana-1747	48	21	)	)	PUNCT
cana-1747	48	22	.	.	PUNCT
cana-1747	49	1	we	we	PRON
cana-1747	49	2	say	say	VERB
cana-1747	49	3	𝐴𝑛	𝐴𝑛	PROPN
cana-1747	49	4	happens	happen	VERB
cana-1747	49	5	infinitely	infinitely	ADV
cana-1747	49	6	often	often	ADV
cana-1747	49	7	,	,	PUNCT
cana-1747	49	8	abbreviated	abbreviate	VERB
cana-1747	49	9	as	as	ADP
cana-1747	49	10	𝐴𝑛	𝐴𝑛	PROPN
cana-1747	49	11	𝑖.	𝑖.	ADV
cana-1747	49	12	𝑜.	𝑜.	VERB
cana-1747	49	13	,	,	PUNCT
cana-1747	49	14	if	if	SCONJ
cana-1747	49	15	for	for	ADP
cana-1747	49	16	all	all	DET
cana-1747	49	17	𝑛	𝑛	PRON
cana-1747	49	18	there	there	PRON
cana-1747	49	19	is	be	VERB
cana-1747	49	20	𝑚	𝑚	PRON
cana-1747	49	21	≥	≥	NOUN
cana-1747	49	22	𝑛	𝑛	DET
cana-1747	49	23	such	such	ADJ
cana-1747	49	24	that	that	SCONJ
cana-1747	49	25	𝐴𝑚	𝐴𝑚	PROPN
cana-1747	49	26	is	be	AUX
cana-1747	49	27	true	true	ADJ
cana-1747	49	28	.	.	PUNCT
cana-1747	50	1	we	we	PRON
cana-1747	50	2	now	now	ADV
cana-1747	50	3	state	state	VERB
cana-1747	50	4	borelli	borelli	PROPN
cana-1747	50	5	lemma	lemma	PROPN
cana-1747	50	6	of	of	ADP
cana-1747	50	7	both	both	DET
cana-1747	50	8	versions	version	NOUN
cana-1747	50	9	.	.	PUNCT
cana-1747	51	1	please	please	INTJ
cana-1747	51	2	see	see	VERB
cana-1747	51	3	[	[	X
cana-1747	51	4	3	3	X
cana-1747	51	5	]	]	PUNCT
cana-1747	51	6	for	for	ADP
cana-1747	51	7	the	the	DET
cana-1747	51	8	proof	proof	NOUN
cana-1747	51	9	.	.	PUNCT
cana-1747	52	1	lemma	lemma	PROPN
cana-1747	52	2	5	5	NUM
cana-1747	52	3	(	(	PUNCT
cana-1747	52	4	borel	borel	PROPN
cana-1747	52	5	-	-	PUNCT
cana-1747	52	6	cantelli	cantelli	PROPN
cana-1747	52	7	1	1	NUM
cana-1747	52	8	)	)	PUNCT
cana-1747	52	9	if	if	SCONJ
cana-1747	52	10	{	{	PUNCT
cana-1747	52	11	𝐴𝑘	𝐴𝑘	NOUN
cana-1747	52	12	}	}	PUNCT
cana-1747	52	13	satisfies	satisfie	NOUN
cana-1747	52	14	∑	∑	PUNCT
cana-1747	52	15	|𝐴𝑘|	|𝐴𝑘|	PROPN
cana-1747	52	16	<	<	X
cana-1747	52	17	∞	∞	PROPN
cana-1747	52	18	,	,	PUNCT
cana-1747	52	19	∞	∞	NUM
cana-1747	52	20	𝑘=1	𝑘=1	PROPN
cana-1747	52	21	then	then	ADV
cana-1747	52	22	|{𝐴𝑘	|{𝐴𝑘	ADP
cana-1747	52	23	𝑖.	𝑖.	ADJ
cana-1747	52	24	𝑜.	𝑜.	NOUN
cana-1747	52	25	}	}	PUNCT
cana-1747	53	1	|	|	CCONJ
cana-1747	53	2	=	=	SYM
cana-1747	53	3	0	0	X
cana-1747	53	4	.	.	PUNCT
cana-1747	54	1	lemma	lemma	PROPN
cana-1747	54	2	6	6	NUM
cana-1747	54	3	(	(	PUNCT
cana-1747	54	4	borel	borel	PROPN
cana-1747	54	5	-	-	PUNCT
cana-1747	54	6	cantelli	cantelli	PROPN
cana-1747	54	7	2	2	NUM
cana-1747	54	8	)	)	PUNCT
cana-1747	54	9	if	if	SCONJ
cana-1747	54	10	independent	independent	ADJ
cana-1747	54	11	events	event	NOUN
cana-1747	54	12	{	{	PUNCT
cana-1747	54	13	𝐴𝑘	𝐴𝑘	PROPN
cana-1747	54	14	}	}	PUNCT
cana-1747	54	15	satisfies	satisfie	NOUN
cana-1747	54	16	∑	∑	PUNCT
cana-1747	54	17	|𝐴𝑘|	|𝐴𝑘|	PROPN
cana-1747	54	18	=	=	SYM
cana-1747	54	19	∞	∞	PROPN
cana-1747	54	20	,	,	PUNCT
cana-1747	54	21	∞	∞	NUM
cana-1747	55	1	𝑘=1	𝑘=1	PROPN
cana-1747	56	1	then	then	ADV
cana-1747	56	2	|{𝐴𝑘	|{𝐴𝑘	ADP
cana-1747	56	3	𝑖.	𝑖.	ADJ
cana-1747	56	4	𝑜.	𝑜.	NOUN
cana-1747	56	5	}	}	PUNCT
cana-1747	57	1	|	|	CCONJ
cana-1747	57	2	=	=	SYM
cana-1747	57	3	1	1	X
cana-1747	57	4	.	.	PUNCT
cana-1747	58	1	next	next	ADV
cana-1747	58	2	,	,	PUNCT
cana-1747	58	3	we	we	PRON
cana-1747	58	4	state	state	VERB
cana-1747	58	5	a	a	DET
cana-1747	58	6	result	result	NOUN
cana-1747	58	7	on	on	ADP
cana-1747	58	8	exponential	exponential	ADJ
cana-1747	58	9	estimate	estimate	NOUN
cana-1747	58	10	for	for	ADP
cana-1747	58	11	independent	independent	ADJ
cana-1747	58	12	random	random	ADJ
cana-1747	58	13	variables	variable	NOUN
cana-1747	58	14	which	which	PRON
cana-1747	58	15	will	will	AUX
cana-1747	58	16	be	be	AUX
cana-1747	58	17	used	use	VERB
cana-1747	58	18	in	in	ADP
cana-1747	58	19	the	the	DET
cana-1747	58	20	proof	proof	NOUN
cana-1747	58	21	of	of	ADP
cana-1747	58	22	our	our	PRON
cana-1747	58	23	main	main	ADJ
cana-1747	58	24	result	result	NOUN
cana-1747	58	25	.	.	PUNCT
cana-1747	59	1	for	for	ADP
cana-1747	59	2	the	the	DET
cana-1747	59	3	proof	proof	NOUN
cana-1747	59	4	,	,	PUNCT
cana-1747	59	5	please	please	INTJ
cana-1747	59	6	see	see	VERB
cana-1747	59	7	[	[	X
cana-1747	59	8	2	2	NUM
cana-1747	59	9	]	]	PUNCT
cana-1747	59	10	.	.	PUNCT
cana-1747	60	1	theorem	theorem	ADJ
cana-1747	60	2	7	7	PROPN
cana-1747	60	3	.	.	PUNCT
cana-1747	61	1	suppose	suppose	VERB
cana-1747	61	2	{	{	PUNCT
cana-1747	61	3	𝑌𝑘	𝑌𝑘	PROPN
cana-1747	61	4	}	}	PUNCT
cana-1747	61	5	is	be	AUX
cana-1747	61	6	a	a	DET
cana-1747	61	7	sequence	sequence	NOUN
cana-1747	61	8	of	of	ADP
cana-1747	61	9	random	random	ADJ
cana-1747	61	10	variables	variable	NOUN
cana-1747	61	11	on	on	ADP
cana-1747	61	12	sample	sample	NOUN
cana-1747	61	13	spac	spac	NOUN
cana-1747	61	14	(	(	PUNCT
cana-1747	61	15	𝐼	𝐼	PROPN
cana-1747	61	16	=	=	SYM
cana-1747	61	17	(	(	PUNCT
cana-1747	61	18	0,1	0,1	NUM
cana-1747	61	19	)	)	PUNCT
cana-1747	61	20	,	,	PUNCT
cana-1747	61	21	ℬ	ℬ	NOUN
cana-1747	61	22	,	,	PUNCT
cana-1747	61	23	𝜇	𝜇	ADP
cana-1747	61	24	)	)	PUNCT
cana-1747	61	25	,	,	PUNCT
cana-1747	61	26	with	with	ADP
cana-1747	61	27	zero	zero	NUM
cana-1747	61	28	mean	mean	NOUN
cana-1747	61	29	and	and	CCONJ
cana-1747	61	30	variance	variance	NOUN
cana-1747	61	31	𝜎𝑘	𝜎𝑘	NOUN
cana-1747	61	32	2	2	X
cana-1747	61	33	.	.	PUNCT
cana-1747	62	1	let	let	VERB
cana-1747	62	2	𝑆𝑛	𝑆𝑛	PROPN
cana-1747	62	3	=	=	NOUN
cana-1747	62	4	∑	∑	PROPN
cana-1747	62	5	𝑌𝑘	𝑌𝑘	PROPN
cana-1747	62	6	𝑛	𝑛	PROPN
cana-1747	62	7	𝑘=1	𝑘=1	NOUN
cana-1747	62	8	,	,	PUNCT
cana-1747	62	9	𝑠𝑛	𝑠𝑛	NOUN
cana-1747	62	10	2	2	NUM
cana-1747	62	11	=	=	NOUN
cana-1747	62	12	∑	∑	PROPN
cana-1747	62	13	𝜎𝑘	𝜎𝑘	NOUN
cana-1747	62	14	2	2	NUM
cana-1747	62	15	𝑛	𝑛	PRON
cana-1747	62	16	𝑘=1	𝑘=1	NOUN
cana-1747	62	17	𝑎𝑛𝑑	𝑎𝑛𝑑	X
cana-1747	62	18	𝑍𝑛	𝑍𝑛	PROPN
cana-1747	62	19	=	=	SYM
cana-1747	62	20	max	max	PROPN
cana-1747	62	21	k≤n	k≤n	PROPN
cana-1747	62	22	|𝑌𝑘|	|𝑌𝑘|	VERB
cana-1747	62	23	𝑠𝑛	𝑠𝑛	NOUN
cana-1747	62	24	then	then	ADV
cana-1747	62	25	,	,	PUNCT
cana-1747	62	26	for	for	SCONJ
cana-1747	62	27	given	give	VERB
cana-1747	62	28	𝛽	𝛽	PROPN
cana-1747	62	29	>	>	X
cana-1747	62	30	0	0	NUM
cana-1747	62	31	,	,	PUNCT
cana-1747	62	32	if	if	SCONJ
cana-1747	62	33	𝑍𝑛(𝛽	𝑍𝑛(𝛽	NOUN
cana-1747	62	34	)	)	PUNCT
cana-1747	62	35	is	be	AUX
cana-1747	62	36	very	very	ADV
cana-1747	62	37	small	small	ADJ
cana-1747	62	38	and	and	CCONJ
cana-1747	62	39	𝛾	𝛾	NOUN
cana-1747	63	1	=	=	SYM
cana-1747	63	2	𝛾(𝛽	𝛾(𝛽	X
cana-1747	63	3	)	)	PUNCT
cana-1747	63	4	is	be	AUX
cana-1747	63	5	very	very	ADV
cana-1747	63	6	large	large	ADJ
cana-1747	63	7	,	,	PUNCT
cana-1747	63	8	then	then	ADV
cana-1747	63	9	|{𝑡	|{𝑡	VERB
cana-1747	63	10	∈	∈	NOUN
cana-1747	63	11	𝐼	𝐼	ADP
cana-1747	63	12	|𝑆𝑛(𝑡)|	|𝑆𝑛(𝑡)|	NOUN
cana-1747	63	13	𝑠𝑛	𝑠𝑛	NOUN
cana-1747	63	14	>	>	X
cana-1747	63	15	𝛾	𝛾	ADP
cana-1747	63	16	}	}	PUNCT
cana-1747	63	17	|	|	ADV
cana-1747	63	18	>	>	X
cana-1747	63	19	exp(−	exp(−	PROPN
cana-1747	63	20	𝛾2	𝛾2	PROPN
cana-1747	63	21	2	2	NUM
cana-1747	63	22	(	(	PUNCT
cana-1747	63	23	1	1	NUM
cana-1747	63	24	+	+	NUM
cana-1747	63	25	𝛽	𝛽	NOUN
cana-1747	63	26	)	)	PUNCT
cana-1747	63	27	.	.	PUNCT
cana-1747	64	1	following	follow	VERB
cana-1747	64	2	,	,	PUNCT
cana-1747	64	3	we	we	PRON
cana-1747	64	4	present	present	VERB
cana-1747	64	5	a	a	DET
cana-1747	64	6	sub	sub	ADJ
cana-1747	64	7	-	-	ADJ
cana-1747	64	8	gaussian	gaussian	ADJ
cana-1747	64	9	type	type	NOUN
cana-1747	64	10	estimate	estimate	NOUN
cana-1747	64	11	crucial	crucial	ADJ
cana-1747	64	12	to	to	ADP
cana-1747	64	13	proving	prove	VERB
cana-1747	64	14	our	our	PRON
cana-1747	64	15	main	main	ADJ
cana-1747	64	16	result	result	NOUN
cana-1747	64	17	.	.	PUNCT
cana-1747	65	1	for	for	ADP
cana-1747	65	2	a	a	DET
cana-1747	65	3	detailed	detailed	ADJ
cana-1747	65	4	proof	proof	NOUN
cana-1747	65	5	,	,	PUNCT
cana-1747	65	6	refer	refer	VERB
cana-1747	65	7	to	to	ADP
cana-1747	65	8	[	[	X
cana-1747	65	9	11	11	NUM
cana-1747	65	10	]	]	PUNCT
cana-1747	65	11	.	.	PUNCT
cana-1747	66	1	we	we	PRON
cana-1747	66	2	sketch	sketch	VERB
cana-1747	66	3	the	the	DET
cana-1747	66	4	proof	proof	NOUN
cana-1747	66	5	.	.	PUNCT
cana-1747	67	1	communications	communication	NOUN
cana-1747	67	2	on	on	ADP
cana-1747	67	3	applied	apply	VERB
cana-1747	67	4	nonlinear	nonlinear	ADJ
cana-1747	67	5	analysis	analysis	NOUN
cana-1747	67	6	issn	issn	NOUN
cana-1747	67	7	:	:	PUNCT
cana-1747	67	8	1074	1074	NUM
cana-1747	67	9	-	-	PUNCT
cana-1747	67	10	133x	133x	NUM
cana-1747	67	11	vol	vol	NOUN
cana-1747	67	12	32	32	NUM
cana-1747	67	13	no	no	NOUN
cana-1747	67	14	.	.	NOUN
cana-1747	67	15	2	2	NUM
cana-1747	67	16	(	(	PUNCT
cana-1747	67	17	2025	2025	NUM
cana-1747	67	18	)	)	PUNCT
cana-1747	67	19	358	358	NUM
cana-1747	68	1	https://internationalpubls.com	https://internationalpubls.com	X
cana-1747	68	2	lemma	lemma	PROPN
cana-1747	68	3	8	8	X
cana-1747	68	4	.	.	PUNCT
cana-1747	69	1	let	let	VERB
cana-1747	69	2	{	{	PUNCT
cana-1747	69	3	𝑏𝑖	𝑏𝑖	VERB
cana-1747	69	4	}	}	PUNCT
cana-1747	69	5	where	where	SCONJ
cana-1747	69	6	𝑏𝑖	𝑏𝑖	PROPN
cana-1747	69	7	∈	∈	PROPN
cana-1747	69	8	ℝ	ℝ	PROPN
cana-1747	69	9	and	and	CCONJ
cana-1747	69	10	{	{	PUNCT
cana-1747	69	11	𝑢𝑖	𝑢𝑖	NOUN
cana-1747	69	12	}	}	PUNCT
cana-1747	69	13	be	be	AUX
cana-1747	69	14	a	a	DET
cana-1747	69	15	sequence	sequence	NOUN
cana-1747	69	16	of	of	ADP
cana-1747	69	17	signum	signum	ADJ
cana-1747	69	18	functions	function	NOUN
cana-1747	69	19	defined	define	VERB
cana-1747	69	20	by	by	ADP
cana-1747	69	21	𝑢𝑖(𝑡	𝑢𝑖(𝑡	NUM
cana-1747	69	22	)	)	PUNCT
cana-1747	70	1	=	=	VERB
cana-1747	70	2	𝑠𝑔𝑛	𝑠𝑔𝑛	PROPN
cana-1747	70	3	(	(	PUNCT
cana-1747	70	4	sin	sin	NOUN
cana-1747	70	5	2𝑖𝜋𝑡	2𝑖𝜋𝑡	NUM
cana-1747	70	6	)	)	PUNCT
cana-1747	70	7	.	.	PUNCT
cana-1747	71	1	then	then	ADV
cana-1747	71	2	for	for	ADP
cana-1747	71	3	all	all	DET
cana-1747	71	4	𝛼	𝛼	PROPN
cana-1747	71	5	>	>	X
cana-1747	71	6	0	0	PUNCT
cana-1747	71	7	and	and	CCONJ
cana-1747	71	8	for	for	ADP
cana-1747	71	9	a	a	DET
cana-1747	71	10	fixed	fix	VERB
cana-1747	71	11	number	number	NOUN
cana-1747	71	12	𝑛	𝑛	NOUN
cana-1747	71	13	,	,	PUNCT
cana-1747	71	14	we	we	PRON
cana-1747	71	15	have	have	VERB
cana-1747	71	16	|{𝑡	|{𝑡	NOUN
cana-1747	71	17	∈	∈	PROPN
cana-1747	71	18	𝐼	𝐼	NOUN
cana-1747	71	19	:	:	PUNCT
cana-1747	71	20	sup	sup	NOUN
cana-1747	71	21	m≥n	m≥n	ADV
cana-1747	71	22	|∑	|∑	VERB
cana-1747	71	23	𝑏𝑖𝑢𝑖(𝑡	𝑏𝑖𝑢𝑖(𝑡	PROPN
cana-1747	71	24	)	)	PUNCT
cana-1747	71	25	∞	∞	NUM
cana-1747	71	26	𝑖=𝑚+1	𝑖=𝑚+1	ADJ
cana-1747	72	1	|	|	ADV
cana-1747	72	2	>	>	X
cana-1747	72	3	𝛼	𝛼	X
cana-1747	72	4	}	}	PUNCT
cana-1747	72	5	|	|	ADV
cana-1747	72	6	≤	≤	NUM
cana-1747	72	7	12	12	NUM
cana-1747	72	8	exp	exp	NOUN
cana-1747	72	9	(	(	PUNCT
cana-1747	72	10	−𝛼2	−𝛼2	PROPN
cana-1747	72	11	2	2	NUM
cana-1747	72	12	∑	∑	NOUN
cana-1747	72	13	𝑏𝑖	𝑏𝑖	ADP
cana-1747	72	14	2	2	NUM
cana-1747	72	15	∞	∞	NUM
cana-1747	72	16	𝑖=𝑛+1	𝑖=𝑛+1	NOUN
cana-1747	72	17	)	)	PUNCT
cana-1747	72	18	.	.	PUNCT
cana-1747	73	1	proof	proof	NOUN
cana-1747	73	2	:	:	PUNCT
cana-1747	73	3	let	let	VERB
cana-1747	73	4	𝑀	𝑀	PROPN
cana-1747	73	5	≫	≫	PROPN
cana-1747	73	6	𝑛	𝑛	PROPN
cana-1747	73	7	and	and	CCONJ
cana-1747	73	8	we	we	PRON
cana-1747	73	9	write	write	VERB
cana-1747	73	10	𝑔𝑖(𝑡	𝑔𝑖(𝑡	NUM
cana-1747	73	11	)	)	PUNCT
cana-1747	73	12	=	=	PUNCT
cana-1747	74	1	∑	∑	PUNCT
cana-1747	74	2	𝑏𝑘𝑢𝑘(𝑡	𝑏𝑘𝑢𝑘(𝑡	NOUN
cana-1747	74	3	)	)	PUNCT
cana-1747	74	4	.	.	PUNCT
cana-1747	75	1	𝑖	𝑖	X
cana-1747	76	1	𝑘=1	𝑘=1	NOUN
cana-1747	76	2	then	then	ADV
cana-1747	76	3	using	use	VERB
cana-1747	76	4	levy	levy	NOUN
cana-1747	76	5	's	's	PART
cana-1747	76	6	inequality	inequality	NOUN
cana-1747	76	7	,	,	PUNCT
cana-1747	76	8	we	we	PRON
cana-1747	76	9	get	get	VERB
cana-1747	76	10	(	(	PUNCT
cana-1747	76	11	2.1	2.1	NUM
cana-1747	76	12	)	)	PUNCT
cana-1747	76	13	|{𝑡	|{𝑡	NOUN
cana-1747	76	14	∈	∈	NOUN
cana-1747	76	15	𝐼	𝐼	PROPN
cana-1747	76	16	:	:	PUNCT
cana-1747	76	17	max	max	PROPN
cana-1747	76	18	m≥m≥n	m≥m≥n	NOUN
cana-1747	76	19	|𝑔𝑚(𝑡	|𝑔𝑚(𝑡	NUM
cana-1747	76	20	)	)	PUNCT
cana-1747	76	21	−	−	NOUN
cana-1747	76	22	𝑔𝑛(𝑡)|	𝑔𝑛(𝑡)|	X
cana-1747	76	23	>	>	X
cana-1747	76	24	𝛼	𝛼	X
cana-1747	76	25	}	}	PUNCT
cana-1747	76	26	|	|	ADV
cana-1747	76	27	≤	≤	NUM
cana-1747	76	28	|{𝑡	|{𝑡	NOUN
cana-1747	76	29	∈	∈	NOUN
cana-1747	76	30	𝐼	𝐼	PROPN
cana-1747	76	31	:	:	PUNCT
cana-1747	76	32	|𝑔𝑀(𝑡	|𝑔𝑀(𝑡	NOUN
cana-1747	76	33	)	)	PUNCT
cana-1747	76	34	−	−	NOUN
cana-1747	76	35	𝑔𝑛(𝑡)|	𝑔𝑛(𝑡)|	PUNCT
cana-1747	76	36	>	>	X
cana-1747	76	37	𝛼}|	𝛼}|	NOUN
cana-1747	76	38	using	use	VERB
cana-1747	76	39	lemma	lemma	PROPN
cana-1747	76	40	1	1	NUM
cana-1747	76	41	in	in	ADP
cana-1747	76	42	[	[	PUNCT
cana-1747	76	43	11	11	NUM
cana-1747	76	44	]	]	PUNCT
cana-1747	76	45	,	,	PUNCT
cana-1747	76	46	we	we	PRON
cana-1747	76	47	get	get	VERB
cana-1747	76	48	(	(	PUNCT
cana-1747	76	49	2.2	2.2	NUM
cana-1747	76	50	)	)	PUNCT
cana-1747	76	51	|{𝑡	|{𝑡	NOUN
cana-1747	76	52	∈	∈	PROPN
cana-1747	76	53	𝐼:max	𝐼:max	NOUN
cana-1747	76	54	m≥n	m≥n	NOUN
cana-1747	76	55	|𝑔𝑚(𝑡	|𝑔𝑚(𝑡	NUM
cana-1747	76	56	)	)	PUNCT
cana-1747	76	57	−	−	PROPN
cana-1747	76	58	𝑔𝑛(𝑡)|	𝑔𝑛(𝑡)|	X
cana-1747	76	59	>	>	X
cana-1747	76	60	𝛼	𝛼	X
cana-1747	76	61	}	}	PUNCT
cana-1747	76	62	|	|	ADV
cana-1747	76	63	≤	≤	NUM
cana-1747	76	64	6	6	NUM
cana-1747	76	65	exp	exp	NOUN
cana-1747	76	66	(	(	PUNCT
cana-1747	76	67	−𝛼2	−𝛼2	PROPN
cana-1747	76	68	2	2	NUM
cana-1747	76	69	∑	∑	NOUN
cana-1747	76	70	𝑏𝑖	𝑏𝑖	ADP
cana-1747	76	71	2	2	NUM
cana-1747	76	72	∞	∞	NUM
cana-1747	76	73	𝑖=𝑛+1	𝑖=𝑛+1	NOUN
cana-1747	76	74	)	)	PUNCT
cana-1747	76	75	using	use	VERB
cana-1747	76	76	(	(	PUNCT
cana-1747	76	77	2.1	2.1	NUM
cana-1747	76	78	)	)	PUNCT
cana-1747	76	79	in	in	ADP
cana-1747	76	80	(	(	PUNCT
cana-1747	76	81	2.2	2.2	NUM
cana-1747	76	82	)	)	PUNCT
cana-1747	76	83	,	,	PUNCT
cana-1747	76	84	we	we	PRON
cana-1747	76	85	get	get	VERB
cana-1747	76	86	|{𝑡	|{𝑡	NOUN
cana-1747	76	87	∈	∈	NOUN
cana-1747	76	88	𝐼	𝐼	NOUN
cana-1747	76	89	:	:	PUNCT
cana-1747	76	90	sup	sup	NOUN
cana-1747	76	91	m≥m≥n	m≥m≥n	X
cana-1747	76	92	|𝑔𝑀(𝑡	|𝑔𝑀(𝑡	NOUN
cana-1747	76	93	)	)	PUNCT
cana-1747	76	94	−	−	ADP
cana-1747	76	95	𝑔𝑚(𝑡)|	𝑔𝑚(𝑡)|	PUNCT
cana-1747	76	96	>	>	X
cana-1747	76	97	𝛼	𝛼	X
cana-1747	76	98	}	}	PUNCT
cana-1747	76	99	|	|	ADV
cana-1747	76	100	≤	≤	NUM
cana-1747	76	101	12	12	NUM
cana-1747	76	102	exp	exp	NOUN
cana-1747	76	103	(	(	PUNCT
cana-1747	76	104	−𝛼2	−𝛼2	PROPN
cana-1747	76	105	2	2	NUM
cana-1747	76	106	∑	∑	NOUN
cana-1747	76	107	𝑏𝑖	𝑏𝑖	ADP
cana-1747	76	108	2	2	NUM
cana-1747	76	109	∞	∞	NUM
cana-1747	76	110	𝑖=𝑛+1	𝑖=𝑛+1	NOUN
cana-1747	76	111	)	)	PUNCT
cana-1747	76	112	.	.	PUNCT
cana-1747	77	1	then	then	ADV
cana-1747	77	2	continuity	continuity	NOUN
cana-1747	77	3	property	property	NOUN
cana-1747	77	4	gives	give	VERB
cana-1747	77	5	|{𝑡	|{𝑡	NOUN
cana-1747	77	6	∈	∈	NOUN
cana-1747	77	7	𝐼	𝐼	NOUN
cana-1747	77	8	:	:	PUNCT
cana-1747	77	9	sup	sup	NOUN
cana-1747	77	10	m≥n	m≥n	NOUN
cana-1747	77	11	|𝑔(𝑡	|𝑔(𝑡	PROPN
cana-1747	77	12	)	)	PUNCT
cana-1747	77	13	−	−	PROPN
cana-1747	78	1	𝑔𝑚(𝑡)|	𝑔𝑚(𝑡)|	PUNCT
cana-1747	78	2	>	>	X
cana-1747	78	3	𝛼	𝛼	X
cana-1747	78	4	}	}	PUNCT
cana-1747	78	5	|	|	ADV
cana-1747	78	6	≤	≤	NUM
cana-1747	78	7	lim	lim	PROPN
cana-1747	78	8	m→∞	m→∞	NUM
cana-1747	78	9	|{𝑡	|{𝑡	NOUN
cana-1747	78	10	∈	∈	NOUN
cana-1747	78	11	𝐼	𝐼	NOUN
cana-1747	78	12	:	:	PUNCT
cana-1747	78	13	sup	sup	NOUN
cana-1747	78	14	m≥m≥n	m≥m≥n	X
cana-1747	78	15	|𝑔𝑀(𝑡	|𝑔𝑀(𝑡	NOUN
cana-1747	78	16	)	)	PUNCT
cana-1747	79	1	−	−	ADP
cana-1747	79	2	𝑔𝑚(𝑡)|	𝑔𝑚(𝑡)|	PUNCT
cana-1747	79	3	>	>	X
cana-1747	79	4	𝛼	𝛼	X
cana-1747	79	5	}	}	PUNCT
cana-1747	79	6	|	|	ADV
cana-1747	79	7	≤	≤	NUM
cana-1747	79	8	12	12	NUM
cana-1747	79	9	exp	exp	NOUN
cana-1747	79	10	(	(	PUNCT
cana-1747	79	11	−𝛼2	−𝛼2	PROPN
cana-1747	79	12	2	2	NUM
cana-1747	79	13	∑	∑	NOUN
cana-1747	79	14	𝑏𝑖	𝑏𝑖	ADP
cana-1747	79	15	2	2	NUM
cana-1747	79	16	∞	∞	NUM
cana-1747	79	17	𝑖=𝑛+1	𝑖=𝑛+1	NOUN
cana-1747	79	18	)	)	PUNCT
cana-1747	79	19	.	.	PUNCT
cana-1747	80	1	thus	thus	ADV
cana-1747	80	2	we	we	PRON
cana-1747	80	3	have	have	VERB
cana-1747	80	4	|{𝑡	|{𝑡	NOUN
cana-1747	80	5	∈	∈	PROPN
cana-1747	80	6	𝐼	𝐼	NOUN
cana-1747	80	7	:	:	PUNCT
cana-1747	80	8	sup	sup	NOUN
cana-1747	80	9	m≥n	m≥n	ADV
cana-1747	80	10	|∑	|∑	VERB
cana-1747	80	11	𝑏𝑖𝑢𝑖(𝑡	𝑏𝑖𝑢𝑖(𝑡	PROPN
cana-1747	80	12	)	)	PUNCT
cana-1747	80	13	∞	∞	NUM
cana-1747	80	14	𝑖=𝑚+1	𝑖=𝑚+1	ADJ
cana-1747	81	1	|	|	ADV
cana-1747	81	2	>	>	X
cana-1747	81	3	𝛼	𝛼	X
cana-1747	81	4	}	}	PUNCT
cana-1747	81	5	|	|	ADV
cana-1747	81	6	≤	≤	NUM
cana-1747	81	7	12	12	NUM
cana-1747	81	8	exp	exp	NOUN
cana-1747	81	9	(	(	PUNCT
cana-1747	81	10	−𝛼2	−𝛼2	PROPN
cana-1747	81	11	2	2	NUM
cana-1747	81	12	∑	∑	NOUN
cana-1747	81	13	𝑏𝑖	𝑏𝑖	ADP
cana-1747	81	14	2	2	NUM
cana-1747	81	15	∞	∞	NUM
cana-1747	81	16	𝑖=𝑛+1	𝑖=𝑛+1	NOUN
cana-1747	81	17	)	)	PUNCT
cana-1747	81	18	.	.	PUNCT
cana-1747	82	1	3	3	X
cana-1747	82	2	.	.	X
cana-1747	82	3	main	main	ADJ
cana-1747	82	4	result	result	NOUN
cana-1747	82	5	:	:	PUNCT
cana-1747	82	6	proof	proof	NOUN
cana-1747	82	7	of	of	ADP
cana-1747	82	8	theorem	theorem	ADJ
cana-1747	82	9	4	4	NUM
cana-1747	82	10	let	let	VERB
cana-1747	82	11	𝜃	𝜃	PRON
cana-1747	82	12	be	be	AUX
cana-1747	82	13	very	very	ADV
cana-1747	82	14	large	large	ADJ
cana-1747	82	15	and	and	CCONJ
cana-1747	82	16	0	0	NUM
cana-1747	82	17	<	<	X
cana-1747	82	18	<	<	X
cana-1747	82	19	𝜖	𝜖	X
cana-1747	82	20	<	<	X
cana-1747	82	21	1	1	NUM
cana-1747	82	22	.	.	PUNCT
cana-1747	83	1	we	we	PRON
cana-1747	83	2	next	next	ADV
cana-1747	83	3	choose	choose	VERB
cana-1747	83	4	0	0	PUNCT
cana-1747	83	5	<	<	X
cana-1747	83	6	𝛼	𝛼	X
cana-1747	83	7	<	<	X
cana-1747	83	8	2	2	NUM
cana-1747	83	9	in	in	ADP
cana-1747	83	10	such	such	DET
cana-1747	83	11	a	a	DET
cana-1747	83	12	way	way	NOUN
cana-1747	83	13	that	that	SCONJ
cana-1747	83	14	(	(	PUNCT
cana-1747	83	15	1	1	NUM
cana-1747	83	16	−	−	NOUN
cana-1747	83	17	𝜖2)(1	𝜖2)(1	PROPN
cana-1747	83	18	+	+	NUM
cana-1747	83	19	𝛼	𝛼	X
cana-1747	83	20	)	)	PUNCT
cana-1747	83	21	>	>	X
cana-1747	83	22	1	1	X
cana-1747	83	23	.	.	X
cana-1747	83	24	define	define	VERB
cana-1747	83	25	stopping	stopping	NOUN
cana-1747	83	26	times	time	NOUN
cana-1747	83	27	by	by	ADP
cana-1747	83	28	𝑛𝑗	𝑛𝑗	ADP
cana-1747	83	29	=	=	SYM
cana-1747	83	30	min	min	NOUN
cana-1747	83	31	(	(	PUNCT
cana-1747	83	32	𝑛:∑	𝑛:∑	PROPN
cana-1747	83	33	𝑏𝑖	𝑏𝑖	ADP
cana-1747	83	34	2	2	NUM
cana-1747	83	35	∞	∞	NUM
cana-1747	83	36	𝑖=𝑛+1	𝑖=𝑛+1	NOUN
cana-1747	83	37	<	<	X
cana-1747	83	38	1	1	NUM
cana-1747	83	39	𝜃𝑗	𝜃𝑗	NOUN
cana-1747	83	40	)	)	PUNCT
cana-1747	83	41	.	.	PUNCT
cana-1747	84	1	we	we	PRON
cana-1747	84	2	have	have	VERB
cana-1747	84	3	∑	∑	PROPN
cana-1747	84	4	𝑏𝑖	𝑏𝑖	ADP
cana-1747	84	5	2	2	NUM
cana-1747	84	6	∞	∞	NUM
cana-1747	84	7	𝑖=𝑛𝑗	𝑖=𝑛𝑗	PUNCT
cana-1747	85	1	=	=	SYM
cana-1747	85	2	𝑏𝑛𝑗	𝑏𝑛𝑗	ADJ
cana-1747	85	3	2	2	NUM
cana-1747	85	4	+	+	NOUN
cana-1747	85	5	∑	∑	PROPN
cana-1747	85	6	𝑏𝑖	𝑏𝑖	ADP
cana-1747	85	7	2	2	NUM
cana-1747	85	8	∞	∞	NUM
cana-1747	85	9	𝑖=𝑛𝑗+1	𝑖=𝑛𝑗+1	PUNCT
cana-1747	85	10	then	then	ADV
cana-1747	85	11	for	for	ADP
cana-1747	85	12	sufficiently	sufficiently	ADV
cana-1747	85	13	large	large	ADJ
cana-1747	85	14	𝑛𝑗	𝑛𝑗	NOUN
cana-1747	85	15	,	,	PUNCT
cana-1747	85	16	we	we	PRON
cana-1747	85	17	have	have	AUX
cana-1747	85	18	(	(	PUNCT
cana-1747	85	19	3.1	3.1	NUM
cana-1747	85	20	)	)	PUNCT
cana-1747	85	21	(	(	PUNCT
cana-1747	85	22	1	1	NUM
cana-1747	85	23	−	−	NOUN
cana-1747	85	24	𝜖2)∑	𝜖2)∑	PUNCT
cana-1747	85	25	𝑏𝑖	𝑏𝑖	ADP
cana-1747	85	26	2	2	NUM
cana-1747	85	27	∞	∞	NUM
cana-1747	85	28	𝑖=𝑛𝑗	𝑖=𝑛𝑗	PUNCT
cana-1747	86	1	<	<	X
cana-1747	86	2	∑	∑	PROPN
cana-1747	86	3	𝑏𝑖	𝑏𝑖	ADP
cana-1747	86	4	2	2	NUM
cana-1747	86	5	∞	∞	NUM
cana-1747	86	6	𝑖=𝑛𝑗+1	𝑖=𝑛𝑗+1	PUNCT
cana-1747	86	7	<	<	X
cana-1747	86	8	1	1	NUM
cana-1747	86	9	𝜃𝑗	𝜃𝑗	NOUN
cana-1747	86	10	communications	communication	NOUN
cana-1747	86	11	on	on	ADP
cana-1747	86	12	applied	apply	VERB
cana-1747	86	13	nonlinear	nonlinear	ADJ
cana-1747	86	14	analysis	analysis	NOUN
cana-1747	86	15	issn	issn	NOUN
cana-1747	86	16	:	:	PUNCT
cana-1747	86	17	1074	1074	NUM
cana-1747	86	18	-	-	PUNCT
cana-1747	86	19	133x	133x	NUM
cana-1747	86	20	vol	vol	NOUN
cana-1747	86	21	32	32	NUM
cana-1747	86	22	no	no	NOUN
cana-1747	86	23	.	.	NOUN
cana-1747	86	24	2	2	NUM
cana-1747	86	25	(	(	PUNCT
cana-1747	86	26	2025	2025	NUM
cana-1747	86	27	)	)	PUNCT
cana-1747	86	28	359	359	NUM
cana-1747	86	29	https://internationalpubls.com	https://internationalpubls.com	X
cana-1747	86	30	by	by	ADP
cana-1747	86	31	definition	definition	NOUN
cana-1747	86	32	of	of	ADP
cana-1747	86	33	𝑛𝑗	𝑛𝑗	ADP
cana-1747	86	34	,	,	PUNCT
cana-1747	86	35	(	(	PUNCT
cana-1747	86	36	3.2	3.2	NUM
cana-1747	86	37	)	)	PUNCT
cana-1747	86	38	(	(	PUNCT
cana-1747	86	39	1	1	NUM
cana-1747	86	40	−	−	NOUN
cana-1747	86	41	𝜖2	𝜖2	NOUN
cana-1747	86	42	)	)	PUNCT
cana-1747	86	43	1	1	NUM
cana-1747	86	44	𝜃𝑗	𝜃𝑗	ADP
cana-1747	86	45	<	<	X
cana-1747	86	46	(	(	PUNCT
cana-1747	86	47	1	1	NUM
cana-1747	86	48	−	−	NOUN
cana-1747	86	49	𝜖2)∑	𝜖2)∑	PUNCT
cana-1747	86	50	𝑏𝑖	𝑏𝑖	ADP
cana-1747	86	51	2	2	NUM
cana-1747	86	52	∞	∞	NUM
cana-1747	86	53	𝑖=𝑛𝑗	𝑖=𝑛𝑗	PUNCT
cana-1747	87	1	so	so	CCONJ
cana-1747	87	2	from	from	ADP
cana-1747	87	3	(	(	PUNCT
cana-1747	87	4	3.1	3.1	NUM
cana-1747	87	5	)	)	PUNCT
cana-1747	87	6	and	and	CCONJ
cana-1747	87	7	(	(	PUNCT
cana-1747	87	8	3.2	3.2	NUM
cana-1747	87	9	)	)	PUNCT
cana-1747	87	10	,	,	PUNCT
cana-1747	87	11	we	we	PRON
cana-1747	87	12	get	get	VERB
cana-1747	87	13	,	,	PUNCT
cana-1747	87	14	(	(	PUNCT
cana-1747	87	15	1	1	NUM
cana-1747	87	16	−	−	NOUN
cana-1747	87	17	𝜖2	𝜖2	NOUN
cana-1747	87	18	)	)	PUNCT
cana-1747	87	19	1	1	NUM
cana-1747	87	20	𝜃𝑗	𝜃𝑗	ADP
cana-1747	87	21	<	<	X
cana-1747	87	22	∑	∑	PROPN
cana-1747	87	23	𝑏𝑖	𝑏𝑖	ADP
cana-1747	87	24	2	2	NUM
cana-1747	87	25	∞	∞	NUM
cana-1747	87	26	𝑖=𝑛𝑗+1	𝑖=𝑛𝑗+1	PUNCT
cana-1747	88	1	<	<	X
cana-1747	88	2	1	1	NUM
cana-1747	88	3	𝜃𝑗	𝜃𝑗	PUNCT
cana-1747	88	4	thus	thus	ADV
cana-1747	88	5	we	we	PRON
cana-1747	88	6	have	have	VERB
cana-1747	88	7	(	(	PUNCT
cana-1747	88	8	3.3	3.3	NUM
cana-1747	88	9	)	)	PUNCT
cana-1747	88	10	∑	∑	ADV
cana-1747	88	11	𝑏𝑖	𝑏𝑖	ADP
cana-1747	88	12	2	2	NUM
cana-1747	88	13	∞	∞	NUM
cana-1747	88	14	𝑖=𝑛𝑗+1	𝑖=𝑛𝑗+1	PUNCT
cana-1747	88	15	∑	∑	PUNCT
cana-1747	88	16	𝑏𝑖	𝑏𝑖	ADP
cana-1747	88	17	2	2	NUM
cana-1747	88	18	∞	∞	NUM
cana-1747	88	19	𝑖=𝑛𝑗+1	𝑖=𝑛𝑗+1	PUNCT
cana-1747	89	1	+	+	PROPN
cana-1747	89	2	1	1	NUM
cana-1747	89	3	≥	≥	NOUN
cana-1747	89	4	(	(	PUNCT
cana-1747	89	5	1	1	NUM
cana-1747	89	6	−	−	PROPN
cana-1747	89	7	𝜖2)𝜃	𝜖2)𝜃	NOUN
cana-1747	89	8	this	this	PRON
cana-1747	89	9	gives	give	VERB
cana-1747	89	10	|{𝑡	|{𝑡	NOUN
cana-1747	89	11	∈	∈	NOUN
cana-1747	89	12	𝐼	𝐼	NOUN
cana-1747	89	13	:	:	PUNCT
cana-1747	89	14	sup	sup	NOUN
cana-1747	89	15	n≥nj+1	n≥nj+1	NOUN
cana-1747	89	16	|∑	|∑	VERB
cana-1747	90	1	𝑏𝑖𝑢𝑖(𝑡	𝑏𝑖𝑢𝑖(𝑡	NOUN
cana-1747	90	2	)	)	PUNCT
cana-1747	91	1	∞	∞	NUM
cana-1747	91	2	𝑖=𝑛+1	𝑖=𝑛+1	NOUN
cana-1747	92	1	|	|	ADV
cana-1747	92	2	>	>	X
cana-1747	93	1	√	√	NUM
cana-1747	93	2	2(1	2(1	NUM
cana-1747	94	1	+	+	PUNCT
cana-1747	94	2	𝛼	𝛼	X
cana-1747	94	3	)	)	PUNCT
cana-1747	94	4	𝜃	𝜃	NOUN
cana-1747	94	5	∑	∑	PROPN
cana-1747	94	6	𝑏𝑖	𝑏𝑖	ADP
cana-1747	94	7	2	2	NUM
cana-1747	94	8	∞	∞	NUM
cana-1747	94	9	𝑖=𝑛𝑗+1	𝑖=𝑛𝑗+1	PUNCT
cana-1747	95	1	ln	ln	ADV
cana-1747	95	2	ln	ln	ADJ
cana-1747	95	3	(	(	PUNCT
cana-1747	95	4	1	1	NUM
cana-1747	95	5	∑	∑	ADV
cana-1747	95	6	𝑏𝑖	𝑏𝑖	ADP
cana-1747	95	7	2	2	NUM
cana-1747	95	8	∞	∞	NUM
cana-1747	95	9	𝑖=𝑛𝑗+1	𝑖=𝑛𝑗+1	PUNCT
cana-1747	95	10	)	)	PUNCT
cana-1747	95	11	}	}	PUNCT
cana-1747	95	12	|	|	ADV
cana-1747	95	13	=	=	SYM
cana-1747	95	14	||	||	NOUN
cana-1747	95	15	{	{	PUNCT
cana-1747	95	16	𝑡	𝑡	PROPN
cana-1747	95	17	∈	∈	PROPN
cana-1747	95	18	𝐼	𝐼	PROPN
cana-1747	95	19	:	:	PUNCT
cana-1747	95	20	sup	sup	NOUN
cana-1747	95	21	n≥nj+1	n≥nj+1	NOUN
cana-1747	95	22	|∑	|∑	VERB
cana-1747	95	23	𝑏𝑖𝑢𝑖(𝑡	𝑏𝑖𝑢𝑖(𝑡	NOUN
cana-1747	95	24	)	)	PUNCT
cana-1747	95	25	∞	∞	NUM
cana-1747	95	26	𝑖=𝑛+1	𝑖=𝑛+1	NOUN
cana-1747	96	1	|	|	ADV
cana-1747	96	2	√∑	√∑	VERB
cana-1747	96	3	𝑏𝑖	𝑏𝑖	ADP
cana-1747	96	4	2	2	NUM
cana-1747	96	5	∞	∞	NUM
cana-1747	96	6	𝑖=𝑛𝑗+1	𝑖=𝑛𝑗+1	PUNCT
cana-1747	97	1	+	+	PROPN
cana-1747	97	2	1	1	NUM
cana-1747	97	3	>	>	SYM
cana-1747	97	4	√	√	NUM
cana-1747	97	5	2(1	2(1	NUM
cana-1747	97	6	+	+	CCONJ
cana-1747	97	7	𝛼)∑	𝛼)∑	VERB
cana-1747	97	8	𝑏𝑖	𝑏𝑖	ADP
cana-1747	97	9	2	2	NUM
cana-1747	97	10	∞	∞	NUM
cana-1747	97	11	𝑖=𝑛𝑗+1	𝑖=𝑛𝑗+1	PUNCT
cana-1747	98	1	𝜃	𝜃	X
cana-1747	98	2	∑	∑	PROPN
cana-1747	98	3	𝑏𝑖	𝑏𝑖	ADP
cana-1747	98	4	2	2	NUM
cana-1747	98	5	∞	∞	NUM
cana-1747	98	6	𝑖=𝑛𝑗+1	𝑖=𝑛𝑗+1	PUNCT
cana-1747	99	1	+	+	PROPN
cana-1747	99	2	1	1	NUM
cana-1747	99	3	ln	ln	NOUN
cana-1747	99	4	ln	ln	NOUN
cana-1747	99	5	(	(	PUNCT
cana-1747	99	6	1	1	NUM
cana-1747	99	7	∑	∑	ADV
cana-1747	99	8	𝑏𝑖	𝑏𝑖	ADP
cana-1747	99	9	2	2	NUM
cana-1747	99	10	∞	∞	NUM
cana-1747	99	11	𝑖=𝑛𝑗+1	𝑖=𝑛𝑗+1	PUNCT
cana-1747	99	12	)	)	PUNCT
cana-1747	99	13	}	}	PUNCT
cana-1747	99	14	||	||	X
cana-1747	100	1	≤	≤	NUM
cana-1747	100	2	||	||	NOUN
cana-1747	100	3	{	{	PUNCT
cana-1747	100	4	𝑡	𝑡	PROPN
cana-1747	100	5	∈	∈	PROPN
cana-1747	100	6	𝐼	𝐼	PROPN
cana-1747	100	7	:	:	PUNCT
cana-1747	100	8	sup	sup	NOUN
cana-1747	100	9	n≥nj+1	n≥nj+1	NOUN
cana-1747	100	10	|∑	|∑	VERB
cana-1747	101	1	𝑏𝑖𝑢𝑖(𝑡	𝑏𝑖𝑢𝑖(𝑡	NOUN
cana-1747	101	2	)	)	PUNCT
cana-1747	102	1	∞	∞	NUM
cana-1747	102	2	𝑖=𝑛+1	𝑖=𝑛+1	NOUN
cana-1747	102	3	|	|	ADV
cana-1747	102	4	√∑	√∑	VERB
cana-1747	102	5	𝑏𝑖	𝑏𝑖	ADP
cana-1747	102	6	2	2	NUM
cana-1747	102	7	∞	∞	NUM
cana-1747	102	8	𝑖=𝑛𝑗+1	𝑖=𝑛𝑗+1	PUNCT
cana-1747	103	1	+	+	PROPN
cana-1747	103	2	1	1	NUM
cana-1747	103	3	√	√	NUM
cana-1747	103	4	2(1	2(1	NUM
cana-1747	103	5	+	+	PUNCT
cana-1747	103	6	𝛼	𝛼	X
cana-1747	103	7	)	)	PUNCT
cana-1747	103	8	𝜃	𝜃	PRON
cana-1747	103	9	𝜃(1	𝜃(1	PUNCT
cana-1747	103	10	−	−	NOUN
cana-1747	103	11	𝜖2	𝜖2	NOUN
cana-1747	103	12	)	)	PUNCT
cana-1747	103	13	ln	ln	PROPN
cana-1747	104	1	ln	ln	ADJ
cana-1747	104	2	(	(	PUNCT
cana-1747	104	3	1	1	NUM
cana-1747	104	4	∑	∑	ADV
cana-1747	104	5	𝑏𝑖	𝑏𝑖	ADP
cana-1747	104	6	2	2	NUM
cana-1747	104	7	∞	∞	NUM
cana-1747	104	8	𝑖=𝑛𝑗+1	𝑖=𝑛𝑗+1	PUNCT
cana-1747	104	9	)	)	PUNCT
cana-1747	104	10	}	}	PUNCT
cana-1747	104	11	||	||	NOUN
cana-1747	105	1	=	=	PUNCT
cana-1747	105	2	|{𝑡	|{𝑡	NOUN
cana-1747	105	3	∈	∈	NOUN
cana-1747	105	4	𝐼	𝐼	NOUN
cana-1747	105	5	:	:	PUNCT
cana-1747	105	6	sup	sup	NOUN
cana-1747	105	7	n≥nj+1	n≥nj+1	NOUN
cana-1747	105	8	|∑	|∑	VERB
cana-1747	105	9	𝑏𝑖𝑢𝑖(𝑡	𝑏𝑖𝑢𝑖(𝑡	NOUN
cana-1747	105	10	)	)	PUNCT
cana-1747	105	11	∞	∞	NUM
cana-1747	105	12	𝑖=𝑛+1	𝑖=𝑛+1	NOUN
cana-1747	106	1	|	|	ADV
cana-1747	106	2	>	>	X
cana-1747	107	1	√	√	NUM
cana-1747	107	2	2(1	2(1	NUM
cana-1747	108	1	+	+	CCONJ
cana-1747	108	2	𝛼)(1	𝛼)(1	ADP
cana-1747	108	3	−	−	PROPN
cana-1747	108	4	𝜖2	𝜖2	NOUN
cana-1747	108	5	)	)	PUNCT
cana-1747	108	6	∑	∑	PUNCT
cana-1747	108	7	𝑏𝑖	𝑏𝑖	ADP
cana-1747	108	8	2	2	NUM
cana-1747	108	9	∞	∞	NUM
cana-1747	108	10	𝑖=𝑛𝑗+1	𝑖=𝑛𝑗+1	PUNCT
cana-1747	109	1	+	+	PROPN
cana-1747	109	2	1	1	NUM
cana-1747	109	3	ln	ln	NOUN
cana-1747	109	4	ln	ln	NOUN
cana-1747	109	5	(	(	PUNCT
cana-1747	109	6	1	1	NUM
cana-1747	109	7	∑	∑	ADV
cana-1747	109	8	𝑏𝑖	𝑏𝑖	ADP
cana-1747	109	9	2	2	NUM
cana-1747	109	10	∞	∞	NUM
cana-1747	109	11	𝑖=𝑛𝑗+1	𝑖=𝑛𝑗+1	PUNCT
cana-1747	109	12	)	)	PUNCT
cana-1747	109	13	}	}	PUNCT
cana-1747	110	1	|	|	NOUN
cana-1747	110	2	=	=	SYM
cana-1747	110	3	|{𝑡	|{𝑡	NOUN
cana-1747	110	4	∈	∈	NOUN
cana-1747	110	5	𝐼	𝐼	NOUN
cana-1747	110	6	:	:	PUNCT
cana-1747	110	7	sup	sup	NOUN
cana-1747	110	8	n≥nj+1	n≥nj+1	NOUN
cana-1747	110	9	|∑	|∑	VERB
cana-1747	110	10	𝑏𝑖𝑢𝑖(𝑡	𝑏𝑖𝑢𝑖(𝑡	NOUN
cana-1747	110	11	)	)	PUNCT
cana-1747	110	12	−∑	−∑	PROPN
cana-1747	110	13	𝑏𝑖𝑢𝑖(𝑡	𝑏𝑖𝑢𝑖(𝑡	PROPN
cana-1747	110	14	)	)	PUNCT
cana-1747	110	15	𝑛	𝑛	PROPN
cana-1747	110	16	𝑖=1	𝑖=1	PUNCT
cana-1747	110	17	∞	∞	NUM
cana-1747	110	18	𝑖=1	𝑖=1	PUNCT
cana-1747	111	1	|	|	ADV
cana-1747	111	2	>	>	X
cana-1747	112	1	√	√	PROPN
cana-1747	112	2	2(1	2(1	NUM
cana-1747	113	1	+	+	CCONJ
cana-1747	113	2	𝛼)(1	𝛼)(1	ADP
cana-1747	113	3	−	−	PROPN
cana-1747	113	4	𝜖2	𝜖2	NOUN
cana-1747	113	5	)	)	PUNCT
cana-1747	113	6	∑	∑	PUNCT
cana-1747	113	7	𝑏𝑖	𝑏𝑖	ADP
cana-1747	113	8	2	2	NUM
cana-1747	113	9	∞	∞	NUM
cana-1747	113	10	𝑖=𝑛𝑗+1	𝑖=𝑛𝑗+1	PUNCT
cana-1747	114	1	+	+	PROPN
cana-1747	114	2	1	1	NUM
cana-1747	114	3	ln	ln	NOUN
cana-1747	114	4	ln	ln	NOUN
cana-1747	114	5	(	(	PUNCT
cana-1747	114	6	1	1	NUM
cana-1747	114	7	∑	∑	ADV
cana-1747	114	8	𝑏𝑖	𝑏𝑖	ADP
cana-1747	114	9	2	2	NUM
cana-1747	114	10	∞	∞	NUM
cana-1747	114	11	𝑖=𝑛𝑗+1	𝑖=𝑛𝑗+1	PUNCT
cana-1747	114	12	)	)	PUNCT
cana-1747	114	13	}	}	PUNCT
cana-1747	114	14	|	|	ADV
cana-1747	114	15	using	use	VERB
cana-1747	114	16	lemma	lemma	PROPN
cana-1747	114	17	8	8	NUM
cana-1747	114	18	,	,	PUNCT
cana-1747	114	19	we	we	PRON
cana-1747	114	20	have	have	VERB
cana-1747	114	21	communications	communication	NOUN
cana-1747	114	22	on	on	ADP
cana-1747	114	23	applied	apply	VERB
cana-1747	114	24	nonlinear	nonlinear	ADJ
cana-1747	114	25	analysis	analysis	NOUN
cana-1747	114	26	issn	issn	NOUN
cana-1747	114	27	:	:	PUNCT
cana-1747	114	28	1074	1074	NUM
cana-1747	114	29	-	-	PUNCT
cana-1747	114	30	133x	133x	NUM
cana-1747	114	31	vol	vol	NOUN
cana-1747	114	32	32	32	NUM
cana-1747	115	1	no	no	NOUN
cana-1747	115	2	.	.	NOUN
cana-1747	115	3	2	2	NUM
cana-1747	115	4	(	(	PUNCT
cana-1747	115	5	2025	2025	NUM
cana-1747	115	6	)	)	PUNCT
cana-1747	116	1	360	360	NUM
cana-1747	116	2	https://internationalpubls.com	https://internationalpubls.com	X
cana-1747	116	3	|{𝑡	|{𝑡	X
cana-1747	116	4	∈	∈	NOUN
cana-1747	116	5	𝐼	𝐼	NOUN
cana-1747	116	6	:	:	PUNCT
cana-1747	116	7	sup	sup	NOUN
cana-1747	116	8	n≥nj+1	n≥nj+1	NOUN
cana-1747	116	9	|∑	|∑	VERB
cana-1747	116	10	𝑏𝑖𝑢𝑖(𝑡	𝑏𝑖𝑢𝑖(𝑡	NOUN
cana-1747	116	11	)	)	PUNCT
cana-1747	116	12	−∑	−∑	PROPN
cana-1747	116	13	𝑏𝑖𝑢𝑖(𝑡	𝑏𝑖𝑢𝑖(𝑡	PROPN
cana-1747	116	14	)	)	PUNCT
cana-1747	116	15	𝑛	𝑛	PROPN
cana-1747	116	16	𝑖=1	𝑖=1	PUNCT
cana-1747	116	17	∞	∞	NUM
cana-1747	116	18	𝑖=1	𝑖=1	PUNCT
cana-1747	117	1	|	|	ADV
cana-1747	117	2	>	>	X
cana-1747	118	1	√	√	PROPN
cana-1747	118	2	2(1	2(1	NUM
cana-1747	119	1	+	+	CCONJ
cana-1747	119	2	𝛼)(1	𝛼)(1	ADP
cana-1747	119	3	−	−	PROPN
cana-1747	119	4	𝜖2	𝜖2	NOUN
cana-1747	119	5	)	)	PUNCT
cana-1747	119	6	∑	∑	PUNCT
cana-1747	119	7	𝑏𝑖	𝑏𝑖	ADP
cana-1747	119	8	2	2	NUM
cana-1747	119	9	∞	∞	NUM
cana-1747	119	10	𝑖=𝑛𝑗+1	𝑖=𝑛𝑗+1	PUNCT
cana-1747	120	1	+	+	PROPN
cana-1747	120	2	1	1	NUM
cana-1747	120	3	ln	ln	NOUN
cana-1747	120	4	ln	ln	NOUN
cana-1747	120	5	(	(	PUNCT
cana-1747	120	6	1	1	NUM
cana-1747	120	7	∑	∑	ADV
cana-1747	120	8	𝑏𝑖	𝑏𝑖	ADP
cana-1747	120	9	2	2	NUM
cana-1747	120	10	∞	∞	NUM
cana-1747	120	11	𝑖=𝑛𝑗+1	𝑖=𝑛𝑗+1	PUNCT
cana-1747	120	12	)	)	PUNCT
cana-1747	120	13	}	}	PUNCT
cana-1747	120	14	|	|	ADV
cana-1747	120	15	≤	≤	NUM
cana-1747	120	16	24	24	NUM
cana-1747	120	17	exp	exp	NOUN
cana-1747	120	18	(	(	PUNCT
cana-1747	120	19	2(1	2(1	NUM
cana-1747	120	20	+	+	CCONJ
cana-1747	120	21	𝛼)(1	𝛼)(1	ADP
cana-1747	120	22	−	−	PROPN
cana-1747	120	23	𝜖2	𝜖2	NOUN
cana-1747	120	24	)	)	PUNCT
cana-1747	120	25	∑	∑	PUNCT
cana-1747	120	26	𝑏𝑖	𝑏𝑖	ADP
cana-1747	120	27	2	2	NUM
cana-1747	120	28	∞	∞	NUM
cana-1747	120	29	𝑖=𝑛𝑗+1	𝑖=𝑛𝑗+1	PUNCT
cana-1747	121	1	+	+	PROPN
cana-1747	121	2	1	1	NUM
cana-1747	121	3	ln	ln	NOUN
cana-1747	121	4	ln	ln	NOUN
cana-1747	121	5	(	(	PUNCT
cana-1747	121	6	1	1	NUM
cana-1747	121	7	∑	∑	ADV
cana-1747	121	8	𝑏𝑖	𝑏𝑖	ADP
cana-1747	121	9	2	2	NUM
cana-1747	121	10	∞	∞	NUM
cana-1747	121	11	𝑖=𝑛𝑗+1	𝑖=𝑛𝑗+1	PUNCT
cana-1747	121	12	)	)	PUNCT
cana-1747	121	13	2	2	NUM
cana-1747	121	14	∑	∑	NOUN
cana-1747	121	15	𝑏𝑖	𝑏𝑖	ADP
cana-1747	121	16	2	2	NUM
cana-1747	121	17	∞	∞	NUM
cana-1747	121	18	𝑖=𝑛𝑗+1	𝑖=𝑛𝑗+1	PUNCT
cana-1747	122	1	+	+	NOUN
cana-1747	122	2	1	1	NUM
cana-1747	122	3	)	)	PUNCT
cana-1747	122	4	=	=	SYM
cana-1747	122	5	24	24	NUM
cana-1747	122	6	(	(	PUNCT
cana-1747	122	7	ln	ln	ADJ
cana-1747	122	8	(	(	PUNCT
cana-1747	122	9	1	1	NUM
cana-1747	122	10	∑	∑	ADV
cana-1747	122	11	𝑏𝑖	𝑏𝑖	ADP
cana-1747	122	12	2	2	NUM
cana-1747	122	13	∞	∞	NUM
cana-1747	122	14	𝑖=𝑛𝑗+1	𝑖=𝑛𝑗+1	PUNCT
cana-1747	122	15	)	)	PUNCT
cana-1747	122	16	)	)	PUNCT
cana-1747	122	17	−(1+𝛼)(1−𝜖2	−(1+𝛼)(1−𝜖2	NOUN
cana-1747	122	18	)	)	PUNCT
cana-1747	122	19	<	<	X
cana-1747	122	20	24	24	NUM
cana-1747	122	21	(	(	PUNCT
cana-1747	122	22	1	1	NUM
cana-1747	122	23	ln	ln	NOUN
cana-1747	122	24	𝜃𝑗	𝜃𝑗	NOUN
cana-1747	122	25	)	)	PUNCT
cana-1747	122	26	(	(	PUNCT
cana-1747	122	27	1+𝛼)(1−𝜖2	1+𝛼)(1−𝜖2	NUM
cana-1747	122	28	)	)	PUNCT
cana-1747	122	29	thus	thus	ADV
cana-1747	122	30	,	,	PUNCT
cana-1747	122	31	|{𝑡	|{𝑡	NOUN
cana-1747	122	32	∈	∈	VERB
cana-1747	122	33	𝐼	𝐼	PROPN
cana-1747	122	34	:	:	PUNCT
cana-1747	122	35	sup	sup	NOUN
cana-1747	123	1	n≥nj+1	n≥nj+1	NOUN
cana-1747	123	2	|∑	|∑	VERB
cana-1747	123	3	𝑏𝑖𝑢𝑖(𝑡	𝑏𝑖𝑢𝑖(𝑡	NOUN
cana-1747	123	4	)	)	PUNCT
cana-1747	123	5	∞	∞	NUM
cana-1747	123	6	𝑖=𝑛+1	𝑖=𝑛+1	NOUN
cana-1747	124	1	|	|	ADV
cana-1747	124	2	>	>	X
cana-1747	125	1	√	√	NUM
cana-1747	125	2	2(1	2(1	NUM
cana-1747	126	1	+	+	PUNCT
cana-1747	126	2	𝛼	𝛼	X
cana-1747	126	3	)	)	PUNCT
cana-1747	126	4	𝜃	𝜃	NOUN
cana-1747	126	5	∑	∑	PROPN
cana-1747	126	6	𝑏𝑖	𝑏𝑖	ADP
cana-1747	126	7	2	2	NUM
cana-1747	126	8	∞	∞	NUM
cana-1747	126	9	𝑖=𝑛𝑗+1	𝑖=𝑛𝑗+1	PUNCT
cana-1747	127	1	ln	ln	ADV
cana-1747	127	2	ln	ln	ADJ
cana-1747	127	3	(	(	PUNCT
cana-1747	127	4	1	1	NUM
cana-1747	127	5	∑	∑	ADV
cana-1747	127	6	𝑏𝑖	𝑏𝑖	ADP
cana-1747	127	7	2	2	NUM
cana-1747	127	8	∞	∞	NUM
cana-1747	127	9	𝑖=𝑛𝑗+1	𝑖=𝑛𝑗+1	PUNCT
cana-1747	127	10	)	)	PUNCT
cana-1747	127	11	}	}	PUNCT
cana-1747	127	12	|	|	CCONJ
cana-1747	127	13	<	<	X
cana-1747	127	14	24	24	NUM
cana-1747	127	15	(	(	PUNCT
cana-1747	127	16	1	1	NUM
cana-1747	127	17	ln	ln	NOUN
cana-1747	127	18	𝜃𝑗	𝜃𝑗	NOUN
cana-1747	127	19	)	)	PUNCT
cana-1747	127	20	(	(	PUNCT
cana-1747	127	21	1+𝛼)(1−𝜖2	1+𝛼)(1−𝜖2	X
cana-1747	127	22	)	)	PUNCT
cana-1747	127	23	define	define	VERB
cana-1747	127	24	𝐴	𝐴	NOUN
cana-1747	127	25	=	=	PUNCT
cana-1747	127	26	{	{	PUNCT
cana-1747	127	27	𝑡	𝑡	NOUN
cana-1747	127	28	∈	∈	PROPN
cana-1747	127	29	𝐼	𝐼	PROPN
cana-1747	127	30	:	:	PUNCT
cana-1747	127	31	sup	sup	NOUN
cana-1747	127	32	n≥nj+1	n≥nj+1	NOUN
cana-1747	127	33	|∑	|∑	VERB
cana-1747	127	34	𝑏𝑖𝑢𝑖(𝑡	𝑏𝑖𝑢𝑖(𝑡	NOUN
cana-1747	127	35	)	)	PUNCT
cana-1747	127	36	∞	∞	NUM
cana-1747	127	37	𝑖=𝑛+1	𝑖=𝑛+1	NOUN
cana-1747	128	1	|	|	ADV
cana-1747	128	2	>	>	X
cana-1747	128	3	√	√	NUM
cana-1747	128	4	2(1+𝛼	2(1+𝛼	NUM
cana-1747	128	5	)	)	PUNCT
cana-1747	128	6	𝜃	𝜃	NOUN
cana-1747	128	7	∑	∑	PROPN
cana-1747	128	8	𝑏𝑖	𝑏𝑖	ADP
cana-1747	128	9	2	2	NUM
cana-1747	128	10	∞	∞	NUM
cana-1747	128	11	𝑖=𝑛𝑗+1	𝑖=𝑛𝑗+1	PUNCT
cana-1747	129	1	ln	ln	ADV
cana-1747	129	2	ln	ln	ADJ
cana-1747	129	3	(	(	PUNCT
cana-1747	129	4	1	1	NUM
cana-1747	129	5	∑	∑	ADV
cana-1747	129	6	𝑏𝑖	𝑏𝑖	ADP
cana-1747	129	7	2	2	NUM
cana-1747	129	8	∞	∞	NUM
cana-1747	129	9	𝑖=𝑛𝑗+1	𝑖=𝑛𝑗+1	PUNCT
cana-1747	129	10	)	)	PUNCT
cana-1747	129	11	}	}	PUNCT
cana-1747	129	12	hence	hence	ADV
cana-1747	129	13	|𝐴|	|𝐴|	VERB
cana-1747	129	14	<	<	X
cana-1747	129	15	24	24	NUM
cana-1747	129	16	(	(	PUNCT
cana-1747	129	17	ln	ln	ADJ
cana-1747	129	18	𝜃)(1+𝛼)(1−𝜖	𝜃)(1+𝛼)(1−𝜖	NOUN
cana-1747	129	19	2	2	NUM
cana-1747	129	20	)	)	PUNCT
cana-1747	129	21	1	1	NUM
cana-1747	129	22	𝑗(1+𝛼)(1−𝜖	𝑗(1+𝛼)(1−𝜖	NOUN
cana-1747	129	23	2	2	NUM
cana-1747	129	24	)	)	PUNCT
cana-1747	129	25	.	.	PUNCT
cana-1747	130	1	set	set	VERB
cana-1747	130	2	𝑆𝑛	𝑆𝑛	PROPN
cana-1747	130	3	=	=	NOUN
cana-1747	130	4	∑	∑	NOUN
cana-1747	130	5	𝑏𝑖𝑢𝑖	𝑏𝑖𝑢𝑖	NOUN
cana-1747	130	6	,	,	PUNCT
cana-1747	130	7	∞	∞	NUM
cana-1747	130	8	𝑖=𝑛+1	𝑖=𝑛+1	ADJ
cana-1747	130	9	𝑠𝑛	𝑠𝑛	NOUN
cana-1747	130	10	2	2	NUM
cana-1747	130	11	=	=	NOUN
cana-1747	130	12	∑	∑	PROPN
cana-1747	130	13	𝑏𝑖	𝑏𝑖	ADP
cana-1747	130	14	2	2	NUM
cana-1747	130	15	𝑛	𝑛	DET
cana-1747	130	16	𝑖=𝑚	𝑖=𝑚	PROPN
cana-1747	130	17	𝑎𝑛𝑑	𝑎𝑛𝑑	ADJ
cana-1747	130	18	𝑍𝑛	𝑍𝑛	PROPN
cana-1747	130	19	=	=	SYM
cana-1747	130	20	max	max	PROPN
cana-1747	130	21	k≤n	k≤n	PROPN
cana-1747	130	22	|𝑏𝑘𝑢𝑘|	|𝑏𝑘𝑢𝑘|	VERB
cana-1747	130	23	𝑠𝑛	𝑠𝑛	NOUN
cana-1747	130	24	fix	fix	NOUN
cana-1747	130	25	𝛽	𝛽	NOUN
cana-1747	130	26	>	>	X
cana-1747	130	27	0	0	PUNCT
cana-1747	130	28	and	and	CCONJ
cana-1747	130	29	choose	choose	VERB
cana-1747	130	30	𝑍𝑛(𝛽	𝑍𝑛(𝛽	NOUN
cana-1747	130	31	)	)	PUNCT
cana-1747	130	32	and	and	CCONJ
cana-1747	130	33	𝛾(𝛽	𝛾(𝛽	PROPN
cana-1747	130	34	)	)	PUNCT
cana-1747	130	35	accordingly	accordingly	ADV
cana-1747	130	36	.	.	PUNCT
cana-1747	131	1	suppose	suppose	VERB
cana-1747	131	2	𝑛𝑗	𝑛𝑗	PART
cana-1747	131	3	is	be	AUX
cana-1747	131	4	sufficiently	sufficiently	ADV
cana-1747	131	5	large	large	ADJ
cana-1747	131	6	.	.	PUNCT
cana-1747	132	1	then	then	ADV
cana-1747	132	2	for	for	ADP
cana-1747	132	3	this	this	PRON
cana-1747	132	4	𝑛𝑗	𝑛𝑗	ADP
cana-1747	132	5	,	,	PUNCT
cana-1747	132	6	theorem	theorem	VERB
cana-1747	132	7	7	7	NUM
cana-1747	132	8	gives	give	VERB
cana-1747	132	9	||	||	NOUN
cana-1747	132	10	{	{	PUNCT
cana-1747	132	11	𝑡	𝑡	NOUN
cana-1747	132	12	∈	∈	NOUN
cana-1747	132	13	𝐼	𝐼	ADP
cana-1747	132	14	∶	∶	NOUN
cana-1747	132	15	|∑	|∑	ADV
cana-1747	132	16	𝑏𝑖𝑢𝑖(𝑡	𝑏𝑖𝑢𝑖(𝑡	NOUN
cana-1747	132	17	)	)	PUNCT
cana-1747	132	18	𝑛	𝑛	PRON
cana-1747	132	19	𝑖=𝑛+1	𝑖=𝑛+1	NOUN
cana-1747	132	20	|	|	ADV
cana-1747	132	21	√∑	√∑	VERB
cana-1747	132	22	𝑏𝑖	𝑏𝑖	ADP
cana-1747	132	23	2	2	NUM
cana-1747	132	24	𝑛	𝑛	NOUN
cana-1747	132	25	𝑖=𝑛𝑗+1	𝑖=𝑛𝑗+1	PUNCT
cana-1747	132	26	>	>	X
cana-1747	132	27	𝛾	𝛾	ADP
cana-1747	132	28	}	}	PUNCT
cana-1747	132	29	||	||	PROPN
cana-1747	132	30	>	>	X
cana-1747	132	31	exp	exp	NOUN
cana-1747	132	32	(	(	PUNCT
cana-1747	132	33	−𝛾2(1	−𝛾2(1	NOUN
cana-1747	132	34	+	+	NUM
cana-1747	132	35	𝛽	𝛽	NOUN
cana-1747	132	36	)	)	PUNCT
cana-1747	132	37	2	2	NUM
cana-1747	132	38	)	)	PUNCT
cana-1747	132	39	.	.	PUNCT
cana-1747	133	1	communications	communication	NOUN
cana-1747	133	2	on	on	ADP
cana-1747	133	3	applied	apply	VERB
cana-1747	133	4	nonlinear	nonlinear	ADJ
cana-1747	133	5	analysis	analysis	NOUN
cana-1747	133	6	issn	issn	NOUN
cana-1747	133	7	:	:	PUNCT
cana-1747	133	8	1074	1074	NUM
cana-1747	133	9	-	-	PUNCT
cana-1747	133	10	133x	133x	NUM
cana-1747	133	11	vol	vol	NOUN
cana-1747	133	12	32	32	NUM
cana-1747	133	13	no	no	NOUN
cana-1747	133	14	.	.	NOUN
cana-1747	133	15	2	2	NUM
cana-1747	133	16	(	(	PUNCT
cana-1747	133	17	2025	2025	NUM
cana-1747	133	18	)	)	PUNCT
cana-1747	133	19	361	361	NUM
cana-1747	133	20	https://internationalpubls.com	https://internationalpubls.com	X
cana-1747	133	21	choose	choose	VERB
cana-1747	133	22	𝛾	𝛾	ADP
cana-1747	133	23	=	=	SYM
cana-1747	133	24	√	√	PROPN
cana-1747	133	25	(	(	PUNCT
cana-1747	133	26	2−𝛼	2−𝛼	NUM
cana-1747	133	27	)	)	PUNCT
cana-1747	133	28	(	(	PUNCT
cana-1747	133	29	1+𝛽	1+𝛽	NUM
cana-1747	133	30	)	)	PUNCT
cana-1747	133	31	ln	ln	NOUN
cana-1747	134	1	ln	ln	ADJ
cana-1747	134	2	(	(	PUNCT
cana-1747	134	3	1	1	NUM
cana-1747	134	4	√∑	√∑	NOUN
cana-1747	134	5	𝑏𝑖	𝑏𝑖	ADP
cana-1747	134	6	2	2	NUM
cana-1747	134	7	∞	∞	NUM
cana-1747	134	8	𝑖=𝑛𝑗+1	𝑖=𝑛𝑗+1	PUNCT
cana-1747	134	9	)	)	PUNCT
cana-1747	134	10	where	where	SCONJ
cana-1747	134	11	𝛼	𝛼	X
cana-1747	134	12	>	>	X
cana-1747	134	13	0	0	X
cana-1747	134	14	.	.	PUNCT
cana-1747	135	1	clearly	clearly	ADV
cana-1747	135	2	for	for	ADP
cana-1747	135	3	large	large	ADJ
cana-1747	135	4	𝑛𝑗	𝑛𝑗	NOUN
cana-1747	135	5	,	,	PUNCT
cana-1747	135	6	𝛾	𝛾	PROPN
cana-1747	135	7	is	be	AUX
cana-1747	135	8	large	large	ADJ
cana-1747	135	9	as	as	SCONJ
cana-1747	135	10	needed	need	VERB
cana-1747	135	11	in	in	ADP
cana-1747	135	12	theorem	theorem	NOUN
cana-1747	135	13	7	7	NUM
cana-1747	135	14	.	.	PUNCT
cana-1747	136	1	thus	thus	ADV
cana-1747	136	2	,	,	PUNCT
cana-1747	136	3	|	|	ADV
cana-1747	136	4	|	|	ADV
cana-1747	136	5	{	{	PUNCT
cana-1747	136	6	𝑡	𝑡	NOUN
cana-1747	136	7	∈	∈	NOUN
cana-1747	136	8	𝐼	𝐼	ADP
cana-1747	136	9	∶	∶	NOUN
cana-1747	136	10	|∑	|∑	ADV
cana-1747	136	11	𝑏𝑖𝑢𝑖(𝑡	𝑏𝑖𝑢𝑖(𝑡	NOUN
cana-1747	136	12	)	)	PUNCT
cana-1747	136	13	𝑛	𝑛	PROPN
cana-1747	136	14	𝑖=𝑛𝑗+1	𝑖=𝑛𝑗+1	PUNCT
cana-1747	136	15	|	|	ADV
cana-1747	136	16	√∑	√∑	ADJ
cana-1747	136	17	𝑏𝑖	𝑏𝑖	ADP
cana-1747	136	18	2	2	NUM
cana-1747	136	19	𝑛	𝑛	NOUN
cana-1747	136	20	𝑖=𝑛𝑗+1	𝑖=𝑛𝑗+1	PUNCT
cana-1747	136	21	>	>	X
cana-1747	136	22	√	√	PUNCT
cana-1747	136	23	(	(	PUNCT
cana-1747	136	24	2	2	NUM
cana-1747	136	25	−	−	NOUN
cana-1747	136	26	𝛼	𝛼	NOUN
cana-1747	136	27	)	)	PUNCT
cana-1747	136	28	(	(	PUNCT
cana-1747	136	29	1	1	NUM
cana-1747	136	30	+	+	NUM
cana-1747	136	31	𝛽	𝛽	NOUN
cana-1747	136	32	)	)	PUNCT
cana-1747	136	33	ln	ln	NOUN
cana-1747	137	1	ln	ln	ADJ
cana-1747	137	2	(	(	PUNCT
cana-1747	137	3	1	1	NUM
cana-1747	137	4	√∑	√∑	NOUN
cana-1747	137	5	𝑏𝑖	𝑏𝑖	ADP
cana-1747	137	6	2	2	NUM
cana-1747	137	7	∞	∞	NUM
cana-1747	137	8	𝑖=𝑛𝑗+1	𝑖=𝑛𝑗+1	PUNCT
cana-1747	137	9	)	)	PUNCT
cana-1747	137	10	}	}	PUNCT
cana-1747	138	1	|	|	ADV
cana-1747	138	2	|	|	ADV
cana-1747	138	3	>	>	X
cana-1747	138	4	exp	exp	PROPN
cana-1747	138	5	(	(	PUNCT
cana-1747	138	6	−(2	−(2	PROPN
cana-1747	138	7	−	−	NOUN
cana-1747	138	8	𝛼	𝛼	NOUN
cana-1747	138	9	)	)	PUNCT
cana-1747	138	10	(	(	PUNCT
cana-1747	138	11	1	1	NUM
cana-1747	138	12	+	+	NUM
cana-1747	138	13	𝛽	𝛽	NOUN
cana-1747	138	14	)	)	PUNCT
cana-1747	138	15	ln	ln	NOUN
cana-1747	138	16	ln	ln	ADJ
cana-1747	138	17	(	(	PUNCT
cana-1747	138	18	1	1	NUM
cana-1747	138	19	√∑	√∑	NOUN
cana-1747	138	20	𝑏𝑖	𝑏𝑖	ADP
cana-1747	138	21	2	2	NUM
cana-1747	138	22	∞	∞	NUM
cana-1747	138	23	𝑖=𝑛𝑗+1	𝑖=𝑛𝑗+1	PUNCT
cana-1747	138	24	)	)	PUNCT
cana-1747	139	1	(	(	PUNCT
cana-1747	139	2	1	1	NUM
cana-1747	139	3	+	+	NUM
cana-1747	139	4	𝛽	𝛽	NOUN
cana-1747	139	5	)	)	PUNCT
cana-1747	139	6	2	2	NUM
cana-1747	139	7	)	)	PUNCT
cana-1747	139	8	≥	≥	NOUN
cana-1747	139	9	1	1	NUM
cana-1747	139	10	(	(	PUNCT
cana-1747	139	11	𝑗	𝑗	PROPN
cana-1747	139	12	ln	ln	NOUN
cana-1747	139	13	𝜃	𝜃	NOUN
cana-1747	140	1	+	+	NOUN
cana-1747	140	2	ln(1	ln(1	ADP
cana-1747	140	3	−	−	NOUN
cana-1747	140	4	𝜖2	𝜖2	NOUN
cana-1747	140	5	)	)	PUNCT
cana-1747	140	6	)	)	PUNCT
cana-1747	141	1	2−𝛼	2−𝛼	NUM
cana-1747	141	2	2	2	NUM
cana-1747	141	3	therefore	therefore	ADV
cana-1747	141	4	for	for	ADP
cana-1747	141	5	large	large	ADJ
cana-1747	141	6	𝑛𝑗	𝑛𝑗	NOUN
cana-1747	141	7	,	,	PUNCT
cana-1747	141	8	we	we	PRON
cana-1747	141	9	have	have	VERB
cana-1747	141	10	|	|	ADV
cana-1747	142	1	|	|	ADV
cana-1747	142	2	|	|	ADV
cana-1747	142	3	{	{	PUNCT
cana-1747	142	4	𝑡	𝑡	NOUN
cana-1747	142	5	∈	∈	NOUN
cana-1747	142	6	𝐼	𝐼	ADP
cana-1747	142	7	∶	∶	NOUN
cana-1747	142	8	|∑	|∑	ADV
cana-1747	143	1	𝑏𝑖𝑢𝑖(𝑡	𝑏𝑖𝑢𝑖(𝑡	NOUN
cana-1747	143	2	)	)	PUNCT
cana-1747	143	3	𝑛	𝑛	PROPN
cana-1747	143	4	𝑖=𝑛𝑗+1	𝑖=𝑛𝑗+1	PUNCT
cana-1747	144	1	|	|	ADV
cana-1747	144	2	√∑	√∑	VERB
cana-1747	144	3	𝑏𝑖	𝑏𝑖	ADP
cana-1747	144	4	2	2	NUM
cana-1747	144	5	(	(	PUNCT
cana-1747	144	6	2−𝛼	2−𝛼	NUM
cana-1747	144	7	)	)	PUNCT
cana-1747	144	8	(	(	PUNCT
cana-1747	144	9	1+𝛽	1+𝛽	NUM
cana-1747	144	10	)	)	PUNCT
cana-1747	144	11	ln	ln	NOUN
cana-1747	145	1	ln	ln	ADJ
cana-1747	146	1	(	(	PUNCT
cana-1747	146	2	1	1	NUM
cana-1747	146	3	√∑	√∑	NOUN
cana-1747	146	4	𝑏𝑖	𝑏𝑖	ADP
cana-1747	146	5	2	2	NUM
cana-1747	146	6	∞	∞	NUM
cana-1747	146	7	𝑖=𝑛𝑗+1	𝑖=𝑛𝑗+1	PUNCT
cana-1747	146	8	)	)	PUNCT
cana-1747	147	1	𝑛	𝑛	ADP
cana-1747	147	2	𝑖=𝑛𝑗+1	𝑖=𝑛𝑗+1	PUNCT
cana-1747	147	3	>	>	X
cana-1747	147	4	1	1	X
cana-1747	147	5	}	}	PUNCT
cana-1747	148	1	|	|	ADV
cana-1747	148	2	|	|	ADV
cana-1747	149	1	|	|	ADV
cana-1747	149	2	>	>	X
cana-1747	149	3	1	1	NUM
cana-1747	149	4	2	2	NUM
cana-1747	149	5	1	1	NUM
cana-1747	149	6	(	(	PUNCT
cana-1747	149	7	𝑗	𝑗	PROPN
cana-1747	149	8	ln	ln	PROPN
cana-1747	149	9	𝜃	𝜃	PROPN
cana-1747	149	10	)	)	PUNCT
cana-1747	149	11	2−𝛼	2−𝛼	NUM
cana-1747	149	12	2	2	NUM
cana-1747	149	13	this	this	PRON
cana-1747	149	14	gives	give	VERB
cana-1747	149	15	|	|	ADV
cana-1747	150	1	|	|	ADV
cana-1747	150	2	|	|	ADV
cana-1747	150	3	{	{	PUNCT
cana-1747	150	4	𝑡	𝑡	NOUN
cana-1747	150	5	∈	∈	NOUN
cana-1747	150	6	𝐼	𝐼	ADP
cana-1747	150	7	∶	∶	NOUN
cana-1747	150	8	|∑	|∑	ADV
cana-1747	151	1	𝑏𝑖𝑢𝑖(𝑡	𝑏𝑖𝑢𝑖(𝑡	NOUN
cana-1747	151	2	)	)	PUNCT
cana-1747	151	3	∞	∞	NUM
cana-1747	151	4	𝑖=𝑛𝑗+1	𝑖=𝑛𝑗+1	PUNCT
cana-1747	152	1	−	−	X
cana-1747	152	2	∑	∑	PUNCT
cana-1747	152	3	𝑏𝑖𝑢𝑖(𝑡	𝑏𝑖𝑢𝑖(𝑡	PROPN
cana-1747	152	4	)	)	PUNCT
cana-1747	153	1	∞	∞	PROPN
cana-1747	153	2	𝑖=𝑛+1	𝑖=𝑛+1	NOUN
cana-1747	153	3	|	|	ADV
cana-1747	153	4	√2∑	√2∑	VERB
cana-1747	153	5	𝑏𝑖	𝑏𝑖	ADP
cana-1747	153	6	2	2	NUM
cana-1747	153	7	ln	ln	NOUN
cana-1747	153	8	ln	ln	ADJ
cana-1747	153	9	(	(	PUNCT
cana-1747	153	10	1	1	NUM
cana-1747	153	11	√∑	√∑	NOUN
cana-1747	153	12	𝑏𝑖	𝑏𝑖	ADP
cana-1747	153	13	2	2	NUM
cana-1747	153	14	∞	∞	NUM
cana-1747	153	15	𝑖=𝑛𝑗+1	𝑖=𝑛𝑗+1	PUNCT
cana-1747	153	16	)	)	PUNCT
cana-1747	154	1	𝑛	𝑛	ADP
cana-1747	154	2	𝑖=𝑛𝑗+1	𝑖=𝑛𝑗+1	PUNCT
cana-1747	154	3	>	>	X
cana-1747	154	4	√	√	PUNCT
cana-1747	154	5	(	(	PUNCT
cana-1747	154	6	2	2	NUM
cana-1747	154	7	−	−	NUM
cana-1747	154	8	𝛼	𝛼	NOUN
cana-1747	154	9	)	)	PUNCT
cana-1747	154	10	2(1	2(1	NUM
cana-1747	154	11	+	+	PUNCT
cana-1747	154	12	𝛽	𝛽	NOUN
cana-1747	154	13	)	)	PUNCT
cana-1747	154	14	}	}	PUNCT
cana-1747	155	1	|	|	ADV
cana-1747	155	2	|	|	ADV
cana-1747	155	3	|	|	ADV
cana-1747	155	4	>	>	X
cana-1747	155	5	1	1	NUM
cana-1747	155	6	2	2	NUM
cana-1747	155	7	1	1	NUM
cana-1747	155	8	(	(	PUNCT
cana-1747	155	9	𝑗	𝑗	PROPN
cana-1747	155	10	ln	ln	PROPN
cana-1747	155	11	𝜃	𝜃	PROPN
cana-1747	155	12	)	)	PUNCT
cana-1747	155	13	2−𝛼	2−𝛼	NUM
cana-1747	155	14	2	2	NUM
cana-1747	155	15	using	use	VERB
cana-1747	155	16	(	(	PUNCT
cana-1747	155	17	3.1	3.1	NUM
cana-1747	155	18	)	)	PUNCT
cana-1747	155	19	for	for	ADP
cana-1747	155	20	𝑛	𝑛	PRON
cana-1747	155	21	≥	≥	NOUN
cana-1747	155	22	𝑛𝑗+1	𝑛𝑗+1	NUM
cana-1747	155	23	,	,	PUNCT
cana-1747	155	24	we	we	PRON
cana-1747	155	25	have	have	VERB
cana-1747	155	26	∑	∑	PROPN
cana-1747	155	27	𝑏𝑖	𝑏𝑖	ADP
cana-1747	155	28	2	2	NUM
cana-1747	155	29	𝑛	𝑛	NOUN
cana-1747	155	30	𝑖=𝑛𝑗+1	𝑖=𝑛𝑗+1	PUNCT
cana-1747	156	1	=	=	NOUN
cana-1747	156	2	∑	∑	PROPN
cana-1747	156	3	𝑏𝑖	𝑏𝑖	ADP
cana-1747	156	4	2	2	NUM
cana-1747	156	5	∞	∞	NUM
cana-1747	156	6	𝑖=𝑛𝑗+1	𝑖=𝑛𝑗+1	PUNCT
cana-1747	157	1	−∑	−∑	PROPN
cana-1747	157	2	𝑏𝑖	𝑏𝑖	ADP
cana-1747	157	3	2	2	NUM
cana-1747	157	4	∞	∞	NUM
cana-1747	157	5	𝑖=𝑛+1	𝑖=𝑛+1	NOUN
cana-1747	157	6	≥	≥	NOUN
cana-1747	157	7	(	(	PUNCT
cana-1747	157	8	1	1	NUM
cana-1747	157	9	−	−	NOUN
cana-1747	157	10	𝜖2	𝜖2	NOUN
cana-1747	157	11	)	)	PUNCT
cana-1747	157	12	1	1	NUM
cana-1747	157	13	𝜃𝑗	𝜃𝑗	ADP
cana-1747	157	14	−	−	NUM
cana-1747	157	15	1	1	NUM
cana-1747	157	16	𝜃𝑗+1	𝜃𝑗+1	ADP
cana-1747	157	17	≥∑	≥∑	PROPN
cana-1747	157	18	𝑏𝑖	𝑏𝑖	ADP
cana-1747	157	19	2	2	NUM
cana-1747	157	20	∞	∞	NUM
cana-1747	157	21	𝑖=𝑛𝑗+1	𝑖=𝑛𝑗+1	PUNCT
cana-1747	157	22	(	(	PUNCT
cana-1747	157	23	1	1	NUM
cana-1747	157	24	−	−	NOUN
cana-1747	157	25	𝜖2	𝜖2	NOUN
cana-1747	157	26	−	−	PROPN
cana-1747	157	27	1	1	NUM
cana-1747	157	28	𝜃	𝜃	NOUN
cana-1747	157	29	)	)	PUNCT
cana-1747	157	30	then	then	ADV
cana-1747	157	31	this	this	PRON
cana-1747	157	32	gives	give	VERB
cana-1747	157	33	communications	communication	NOUN
cana-1747	157	34	on	on	ADP
cana-1747	157	35	applied	apply	VERB
cana-1747	157	36	nonlinear	nonlinear	ADJ
cana-1747	157	37	analysis	analysis	NOUN
cana-1747	157	38	issn	issn	NOUN
cana-1747	157	39	:	:	PUNCT
cana-1747	157	40	1074	1074	NUM
cana-1747	157	41	-	-	PUNCT
cana-1747	157	42	133x	133x	NUM
cana-1747	157	43	vol	vol	NOUN
cana-1747	157	44	32	32	NUM
cana-1747	157	45	no	no	NOUN
cana-1747	157	46	.	.	NOUN
cana-1747	157	47	2	2	NUM
cana-1747	157	48	(	(	PUNCT
cana-1747	157	49	2025	2025	NUM
cana-1747	157	50	)	)	PUNCT
cana-1747	157	51	362	362	NUM
cana-1747	157	52	https://internationalpubls.com	https://internationalpubls.com	X
cana-1747	158	1	|	|	ADV
cana-1747	158	2	|	|	ADV
cana-1747	158	3	|	|	ADV
cana-1747	158	4	{	{	PUNCT
cana-1747	158	5	𝑡	𝑡	NOUN
cana-1747	158	6	∈	∈	NOUN
cana-1747	158	7	𝐼	𝐼	ADP
cana-1747	158	8	∶	∶	NOUN
cana-1747	158	9	|∑	|∑	ADV
cana-1747	159	1	𝑏𝑖𝑢𝑖(𝑡	𝑏𝑖𝑢𝑖(𝑡	NOUN
cana-1747	159	2	)	)	PUNCT
cana-1747	159	3	∞	∞	NUM
cana-1747	159	4	𝑖=𝑛𝑗+1	𝑖=𝑛𝑗+1	PUNCT
cana-1747	160	1	−	−	X
cana-1747	160	2	∑	∑	PUNCT
cana-1747	160	3	𝑏𝑖𝑢𝑖(𝑡	𝑏𝑖𝑢𝑖(𝑡	PROPN
cana-1747	160	4	)	)	PUNCT
cana-1747	161	1	∞	∞	PROPN
cana-1747	161	2	𝑖=𝑛+1	𝑖=𝑛+1	NOUN
cana-1747	161	3	|	|	ADV
cana-1747	161	4	√2∑	√2∑	VERB
cana-1747	161	5	𝑏𝑖	𝑏𝑖	ADP
cana-1747	161	6	2	2	NUM
cana-1747	161	7	(	(	PUNCT
cana-1747	161	8	1	1	NUM
cana-1747	161	9	−	−	NOUN
cana-1747	161	10	𝜖2	𝜖2	NOUN
cana-1747	161	11	−	−	PROPN
cana-1747	161	12	1	1	NUM
cana-1747	161	13	𝜃	𝜃	NOUN
cana-1747	161	14	)	)	PUNCT
cana-1747	162	1	ln	ln	PROPN
cana-1747	163	1	ln	ln	ADJ
cana-1747	163	2	(	(	PUNCT
cana-1747	163	3	1	1	NUM
cana-1747	163	4	√∑	√∑	NOUN
cana-1747	163	5	𝑏𝑖	𝑏𝑖	ADP
cana-1747	163	6	2	2	NUM
cana-1747	163	7	∞	∞	NUM
cana-1747	163	8	𝑖=𝑛𝑗+1	𝑖=𝑛𝑗+1	PUNCT
cana-1747	163	9	)	)	PUNCT
cana-1747	164	1	∞	∞	NUM
cana-1747	164	2	𝑖=𝑛𝑗+1	𝑖=𝑛𝑗+1	PUNCT
cana-1747	164	3	>	>	X
cana-1747	164	4	√	√	PUNCT
cana-1747	164	5	(	(	PUNCT
cana-1747	164	6	2	2	NUM
cana-1747	164	7	−	−	NUM
cana-1747	164	8	𝛼	𝛼	NOUN
cana-1747	164	9	)	)	PUNCT
cana-1747	164	10	2(1	2(1	NUM
cana-1747	164	11	+	+	PUNCT
cana-1747	164	12	𝛽	𝛽	NOUN
cana-1747	164	13	)	)	PUNCT
cana-1747	164	14	}	}	PUNCT
cana-1747	165	1	|	|	ADV
cana-1747	165	2	|	|	ADV
cana-1747	165	3	|	|	ADV
cana-1747	165	4	>	>	X
cana-1747	165	5	1	1	NUM
cana-1747	165	6	2	2	NUM
cana-1747	165	7	1	1	NUM
cana-1747	165	8	(	(	PUNCT
cana-1747	165	9	𝑗	𝑗	PROPN
cana-1747	165	10	ln	ln	PROPN
cana-1747	165	11	𝜃	𝜃	PROPN
cana-1747	165	12	)	)	PUNCT
cana-1747	165	13	2−𝛼	2−𝛼	NUM
cana-1747	165	14	2	2	NUM
cana-1747	165	15	i.e.	i.e.	X
cana-1747	165	16	|	|	ADV
cana-1747	165	17	|	|	ADV
cana-1747	166	1	|	|	ADV
cana-1747	166	2	{	{	PUNCT
cana-1747	166	3	𝑡	𝑡	NOUN
cana-1747	166	4	∈	∈	NOUN
cana-1747	166	5	𝐼	𝐼	ADP
cana-1747	166	6	∶	∶	NOUN
cana-1747	166	7	|∑	|∑	ADV
cana-1747	167	1	𝑏𝑖𝑢𝑖(𝑡	𝑏𝑖𝑢𝑖(𝑡	NOUN
cana-1747	167	2	)	)	PUNCT
cana-1747	167	3	∞	∞	NUM
cana-1747	167	4	𝑖=𝑛𝑗+1	𝑖=𝑛𝑗+1	PUNCT
cana-1747	168	1	−	−	X
cana-1747	168	2	∑	∑	PUNCT
cana-1747	168	3	𝑏𝑖𝑢𝑖(𝑡	𝑏𝑖𝑢𝑖(𝑡	PROPN
cana-1747	168	4	)	)	PUNCT
cana-1747	169	1	∞	∞	PROPN
cana-1747	169	2	𝑖=𝑛+1	𝑖=𝑛+1	NOUN
cana-1747	169	3	|	|	ADV
cana-1747	169	4	√2∑	√2∑	VERB
cana-1747	169	5	𝑏𝑖	𝑏𝑖	ADP
cana-1747	169	6	2	2	NUM
cana-1747	169	7	ln	ln	NOUN
cana-1747	169	8	ln	ln	ADJ
cana-1747	169	9	(	(	PUNCT
cana-1747	169	10	1	1	NUM
cana-1747	169	11	√∑	√∑	NOUN
cana-1747	169	12	𝑏𝑖	𝑏𝑖	ADP
cana-1747	169	13	2	2	NUM
cana-1747	169	14	∞	∞	NUM
cana-1747	169	15	𝑖=𝑛𝑗+1	𝑖=𝑛𝑗+1	PUNCT
cana-1747	169	16	)	)	PUNCT
cana-1747	170	1	∞	∞	NUM
cana-1747	170	2	𝑖=𝑛𝑗+1	𝑖=𝑛𝑗+1	PUNCT
cana-1747	170	3	>	>	X
cana-1747	170	4	√	√	PUNCT
cana-1747	170	5	(	(	PUNCT
cana-1747	170	6	2	2	NUM
cana-1747	170	7	−	−	NOUN
cana-1747	170	8	𝛼	𝛼	NOUN
cana-1747	170	9	)	)	PUNCT
cana-1747	170	10	(	(	PUNCT
cana-1747	170	11	1	1	NUM
cana-1747	170	12	−	−	NOUN
cana-1747	170	13	𝜖2	𝜖2	NOUN
cana-1747	170	14	−	−	PROPN
cana-1747	170	15	1	1	NUM
cana-1747	170	16	𝜃	𝜃	NOUN
cana-1747	170	17	)	)	PUNCT
cana-1747	170	18	2(1	2(1	NUM
cana-1747	171	1	+	+	PUNCT
cana-1747	171	2	𝛽	𝛽	NOUN
cana-1747	171	3	)	)	PUNCT
cana-1747	171	4	}	}	PUNCT
cana-1747	172	1	|	|	ADV
cana-1747	172	2	|	|	ADV
cana-1747	172	3	|	|	ADV
cana-1747	172	4	>	>	X
cana-1747	172	5	1	1	NUM
cana-1747	172	6	2	2	NUM
cana-1747	172	7	1	1	NUM
cana-1747	172	8	(	(	PUNCT
cana-1747	172	9	𝑗	𝑗	PROPN
cana-1747	172	10	ln	ln	PROPN
cana-1747	172	11	𝜃	𝜃	PROPN
cana-1747	172	12	)	)	PUNCT
cana-1747	172	13	2−𝛼	2−𝛼	NUM
cana-1747	172	14	2	2	NUM
cana-1747	172	15	define	define	VERB
cana-1747	172	16	𝐵	𝐵	NOUN
cana-1747	172	17	≔	≔	NOUN
cana-1747	172	18	{	{	PUNCT
cana-1747	172	19	𝑡	𝑡	NOUN
cana-1747	172	20	∈	∈	NOUN
cana-1747	172	21	𝐼	𝐼	ADP
cana-1747	172	22	∶	∶	NOUN
cana-1747	172	23	|∑	|∑	ADV
cana-1747	173	1	𝑏𝑖𝑢𝑖(𝑡	𝑏𝑖𝑢𝑖(𝑡	NOUN
cana-1747	173	2	)	)	PUNCT
cana-1747	173	3	∞	∞	NUM
cana-1747	173	4	𝑖=𝑛𝑗+1	𝑖=𝑛𝑗+1	PUNCT
cana-1747	174	1	−	−	X
cana-1747	174	2	∑	∑	PUNCT
cana-1747	174	3	𝑏𝑖𝑢𝑖(𝑡	𝑏𝑖𝑢𝑖(𝑡	PROPN
cana-1747	174	4	)	)	PUNCT
cana-1747	175	1	∞	∞	PROPN
cana-1747	175	2	𝑖=𝑛+1	𝑖=𝑛+1	NOUN
cana-1747	175	3	|	|	ADV
cana-1747	175	4	√2∑	√2∑	VERB
cana-1747	175	5	𝑏𝑖	𝑏𝑖	ADP
cana-1747	175	6	2	2	NUM
cana-1747	175	7	ln	ln	NOUN
cana-1747	175	8	ln	ln	ADJ
cana-1747	175	9	(	(	PUNCT
cana-1747	175	10	1	1	NUM
cana-1747	175	11	√∑	√∑	NOUN
cana-1747	175	12	𝑏𝑖	𝑏𝑖	ADP
cana-1747	175	13	2	2	NUM
cana-1747	175	14	∞	∞	NUM
cana-1747	175	15	𝑖=𝑛𝑗+1	𝑖=𝑛𝑗+1	PUNCT
cana-1747	175	16	)	)	PUNCT
cana-1747	176	1	∞	∞	NUM
cana-1747	176	2	𝑖=𝑛𝑗+1	𝑖=𝑛𝑗+1	PUNCT
cana-1747	176	3	>	>	X
cana-1747	176	4	√	√	PUNCT
cana-1747	176	5	(	(	PUNCT
cana-1747	176	6	2	2	NUM
cana-1747	176	7	−	−	NOUN
cana-1747	176	8	𝛼	𝛼	NOUN
cana-1747	176	9	)	)	PUNCT
cana-1747	176	10	(	(	PUNCT
cana-1747	176	11	1	1	NUM
cana-1747	176	12	−	−	NOUN
cana-1747	176	13	𝜖2	𝜖2	NOUN
cana-1747	176	14	−	−	PROPN
cana-1747	176	15	1	1	NUM
cana-1747	176	16	𝜃	𝜃	NOUN
cana-1747	176	17	)	)	PUNCT
cana-1747	176	18	2(1	2(1	NUM
cana-1747	177	1	+	+	PUNCT
cana-1747	177	2	𝛽	𝛽	NOUN
cana-1747	177	3	)	)	PUNCT
cana-1747	177	4	}	}	PUNCT
cana-1747	177	5	consequently	consequently	ADV
cana-1747	177	6	,	,	PUNCT
cana-1747	177	7	|𝐵|	|𝐵|	X
cana-1747	177	8	≥	≥	NOUN
cana-1747	177	9	1	1	NUM
cana-1747	177	10	2	2	NUM
cana-1747	177	11	1	1	NUM
cana-1747	177	12	(	(	PUNCT
cana-1747	177	13	𝑗	𝑗	INTJ
cana-1747	177	14	ln𝜃	ln𝜃	PROPN
cana-1747	177	15	)	)	PUNCT
cana-1747	177	16	2−𝛼	2−𝛼	NUM
cana-1747	177	17	2	2	NUM
cana-1747	177	18	.	.	PUNCT
cana-1747	178	1	next	next	ADV
cana-1747	178	2	define	define	VERB
cana-1747	178	3	𝐶	𝐶	PROPN
cana-1747	178	4	≔	≔	NOUN
cana-1747	178	5	{	{	PUNCT
cana-1747	178	6	𝑡	𝑡	NOUN
cana-1747	178	7	∈	∈	NOUN
cana-1747	178	8	𝐼	𝐼	ADP
cana-1747	178	9	∶	∶	NOUN
cana-1747	178	10	|∑	|∑	ADV
cana-1747	178	11	𝑏𝑖𝑢𝑖(𝑡	𝑏𝑖𝑢𝑖(𝑡	NOUN
cana-1747	178	12	)	)	PUNCT
cana-1747	178	13	∞	∞	NUM
cana-1747	178	14	𝑖=𝑛𝑗+1	𝑖=𝑛𝑗+1	PUNCT
cana-1747	179	1	+	+	NOUN
cana-1747	179	2	1	1	NUM
cana-1747	179	3	−	−	NOUN
cana-1747	179	4	∑	∑	ADV
cana-1747	179	5	𝑏𝑖𝑢𝑖(𝑡	𝑏𝑖𝑢𝑖(𝑡	PROPN
cana-1747	179	6	)	)	PUNCT
cana-1747	179	7	∞	∞	NUM
cana-1747	179	8	𝑖=𝑛𝑗+1	𝑖=𝑛𝑗+1	PUNCT
cana-1747	180	1	|	|	ADV
cana-1747	180	2	√2∑	√2∑	VERB
cana-1747	180	3	𝑏𝑖	𝑏𝑖	ADP
cana-1747	180	4	2	2	NUM
cana-1747	180	5	ln	ln	NOUN
cana-1747	180	6	ln	ln	ADJ
cana-1747	180	7	(	(	PUNCT
cana-1747	180	8	1	1	NUM
cana-1747	180	9	√∑	√∑	NOUN
cana-1747	180	10	𝑏𝑖	𝑏𝑖	ADP
cana-1747	180	11	2	2	NUM
cana-1747	180	12	∞	∞	NUM
cana-1747	180	13	𝑖=𝑛𝑗+1	𝑖=𝑛𝑗+1	PUNCT
cana-1747	180	14	)	)	PUNCT
cana-1747	181	1	∞	∞	NUM
cana-1747	181	2	𝑖=𝑛𝑗+1	𝑖=𝑛𝑗+1	PUNCT
cana-1747	181	3	>	>	X
cana-1747	181	4	√	√	PUNCT
cana-1747	181	5	(	(	PUNCT
cana-1747	181	6	2	2	NUM
cana-1747	181	7	−	−	NOUN
cana-1747	181	8	𝛼	𝛼	NOUN
cana-1747	181	9	)	)	PUNCT
cana-1747	181	10	(	(	PUNCT
cana-1747	181	11	1	1	NUM
cana-1747	181	12	−	−	NOUN
cana-1747	181	13	𝜖2	𝜖2	NOUN
cana-1747	181	14	−	−	PROPN
cana-1747	181	15	1	1	NUM
cana-1747	181	16	𝜃	𝜃	NOUN
cana-1747	181	17	)	)	PUNCT
cana-1747	181	18	2(1	2(1	NUM
cana-1747	182	1	+	+	PUNCT
cana-1747	182	2	𝛽	𝛽	NOUN
cana-1747	182	3	)	)	PUNCT
cana-1747	182	4	−	−	PROPN
cana-1747	182	5	2√	2√	PROPN
cana-1747	182	6	(	(	PUNCT
cana-1747	182	7	1	1	NUM
cana-1747	182	8	−	−	NOUN
cana-1747	182	9	𝜖2	𝜖2	NOUN
cana-1747	182	10	)	)	PUNCT
cana-1747	182	11	𝜃	𝜃	NOUN
cana-1747	182	12	(	(	PUNCT
cana-1747	182	13	1	1	NUM
cana-1747	182	14	+	+	NUM
cana-1747	182	15	𝛼	𝛼	X
cana-1747	182	16	)	)	PUNCT
cana-1747	182	17	}	}	PUNCT
cana-1747	182	18	using	use	VERB
cana-1747	182	19	triangle	triangle	NOUN
cana-1747	182	20	inequality	inequality	NOUN
cana-1747	182	21	,	,	PUNCT
cana-1747	182	22	we	we	PRON
cana-1747	182	23	have	have	VERB
cana-1747	182	24	𝐵⋂𝐴𝑐	𝐵⋂𝐴𝑐	X
cana-1747	183	1	⊂	⊂	X
cana-1747	183	2	𝐶.	𝐶.	PROPN
cana-1747	184	1	so	so	ADV
cana-1747	184	2	we	we	PRON
cana-1747	184	3	have	have	AUX
cana-1747	184	4	|𝐵	|𝐵	VERB
cana-1747	184	5	−	−	PUNCT
cana-1747	184	6	𝐴|	𝐴|	PROPN
cana-1747	184	7	≤	≤	ADJ
cana-1747	184	8	|𝐶|	|𝐶|	NOUN
cana-1747	184	9	.	.	PUNCT
cana-1747	185	1	thus	thus	ADV
cana-1747	185	2	,	,	PUNCT
cana-1747	185	3	we	we	PRON
cana-1747	185	4	have	have	VERB
cana-1747	185	5	|𝐶|	|𝐶|	NOUN
cana-1747	185	6	≥	≥	NOUN
cana-1747	185	7	1	1	NUM
cana-1747	185	8	2(𝑗	2(𝑗	NUM
cana-1747	185	9	ln	ln	PROPN
cana-1747	185	10	𝜃	𝜃	NOUN
cana-1747	185	11	)	)	PUNCT
cana-1747	185	12	2−𝛼	2−𝛼	NUM
cana-1747	185	13	2	2	NUM
cana-1747	185	14	−	−	NUM
cana-1747	185	15	24	24	NUM
cana-1747	186	1	(	(	PUNCT
cana-1747	186	2	𝑗	𝑗	PROPN
cana-1747	186	3	ln	ln	ADJ
cana-1747	186	4	𝜃)(1−𝜖	𝜃)(1−𝜖	PROPN
cana-1747	186	5	2)(1+𝛼	2)(1+𝛼	NUM
cana-1747	186	6	)	)	PUNCT
cana-1747	186	7	.	.	PUNCT
cana-1747	187	1	since	since	SCONJ
cana-1747	187	2	𝛼	𝛼	PROPN
cana-1747	187	3	∈	∈	PROPN
cana-1747	187	4	(	(	PUNCT
cana-1747	187	5	0	0	NUM
cana-1747	187	6	,	,	PUNCT
cana-1747	187	7	2	2	NUM
cana-1747	187	8	)	)	PUNCT
cana-1747	187	9	and	and	CCONJ
cana-1747	187	10	(	(	PUNCT
cana-1747	187	11	1	1	NUM
cana-1747	187	12	−	−	NOUN
cana-1747	187	13	𝜖2)(1	𝜖2)(1	PROPN
cana-1747	187	14	+	+	NUM
cana-1747	187	15	𝛼	𝛼	X
cana-1747	187	16	)	)	PUNCT
cana-1747	187	17	>	>	X
cana-1747	187	18	1	1	NUM
cana-1747	187	19	,	,	PUNCT
cana-1747	187	20	for	for	ADP
cana-1747	187	21	large	large	ADJ
cana-1747	187	22	𝑗	𝑗	PROPN
cana-1747	187	23	,	,	PUNCT
cana-1747	187	24	communications	communication	NOUN
cana-1747	187	25	on	on	ADP
cana-1747	187	26	applied	apply	VERB
cana-1747	187	27	nonlinear	nonlinear	ADJ
cana-1747	187	28	analysis	analysis	NOUN
cana-1747	187	29	issn	issn	NOUN
cana-1747	187	30	:	:	PUNCT
cana-1747	187	31	1074	1074	NUM
cana-1747	187	32	-	-	PUNCT
cana-1747	187	33	133x	133x	NUM
cana-1747	187	34	vol	vol	NOUN
cana-1747	187	35	32	32	NUM
cana-1747	187	36	no	no	NOUN
cana-1747	187	37	.	.	NOUN
cana-1747	187	38	2	2	NUM
cana-1747	187	39	(	(	PUNCT
cana-1747	187	40	2025	2025	NUM
cana-1747	187	41	)	)	PUNCT
cana-1747	187	42	363	363	NUM
cana-1747	187	43	https://internationalpubls.com	https://internationalpubls.com	X
cana-1747	187	44	1	1	NUM
cana-1747	187	45	3(𝑗	3(𝑗	NUM
cana-1747	187	46	ln	ln	PROPN
cana-1747	187	47	𝜃	𝜃	NOUN
cana-1747	187	48	)	)	PUNCT
cana-1747	187	49	(	(	PUNCT
cana-1747	187	50	2−𝛼	2−𝛼	NUM
cana-1747	187	51	)	)	PUNCT
cana-1747	187	52	2	2	NUM
cana-1747	187	53	≥	≥	NOUN
cana-1747	187	54	24	24	NUM
cana-1747	187	55	(	(	PUNCT
cana-1747	187	56	𝑗	𝑗	PROPN
cana-1747	187	57	ln	ln	ADJ
cana-1747	187	58	𝜃)(1−𝜖	𝜃)(1−𝜖	PROPN
cana-1747	187	59	2)(1+𝛼	2)(1+𝛼	NUM
cana-1747	187	60	)	)	PUNCT
cana-1747	187	61	this	this	PRON
cana-1747	187	62	gives	give	VERB
cana-1747	187	63	|𝐶|	|𝐶|	NOUN
cana-1747	187	64	≥	≥	NOUN
cana-1747	187	65	1	1	NUM
cana-1747	187	66	6(𝑗	6(𝑗	NUM
cana-1747	187	67	ln	ln	NOUN
cana-1747	187	68	𝜃	𝜃	NOUN
cana-1747	187	69	)	)	PUNCT
cana-1747	187	70	(	(	PUNCT
cana-1747	187	71	2−𝛼	2−𝛼	NUM
cana-1747	187	72	)	)	PUNCT
cana-1747	187	73	2	2	NUM
cana-1747	187	74	.	.	PUNCT
cana-1747	188	1	now	now	ADV
cana-1747	188	2	summing	sum	VERB
cana-1747	188	3	over	over	ADP
cana-1747	188	4	all	all	PRON
cana-1747	188	5	𝑗	𝑗	VERB
cana-1747	188	6	we	we	PRON
cana-1747	188	7	have	have	VERB
cana-1747	188	8	,	,	PUNCT
cana-1747	188	9	∑	∑	ADV
cana-1747	188	10	|	|	ADV
cana-1747	189	1	|	|	ADV
cana-1747	189	2	|	|	ADV
cana-1747	189	3	{	{	PUNCT
cana-1747	189	4	𝑡	𝑡	NOUN
cana-1747	189	5	∈	∈	NOUN
cana-1747	189	6	𝐼	𝐼	ADP
cana-1747	189	7	∶	∶	NOUN
cana-1747	189	8	|∑	|∑	ADV
cana-1747	189	9	𝑏𝑖𝑢𝑖(𝑡	𝑏𝑖𝑢𝑖(𝑡	NOUN
cana-1747	189	10	)	)	PUNCT
cana-1747	189	11	∞	∞	NUM
cana-1747	189	12	𝑖=𝑛𝑗+1	𝑖=𝑛𝑗+1	PUNCT
cana-1747	190	1	+	+	NOUN
cana-1747	190	2	1	1	NUM
cana-1747	190	3	−	−	NOUN
cana-1747	190	4	∑	∑	ADV
cana-1747	190	5	𝑏𝑖𝑢𝑖(𝑡	𝑏𝑖𝑢𝑖(𝑡	PROPN
cana-1747	190	6	)	)	PUNCT
cana-1747	190	7	∞	∞	NUM
cana-1747	190	8	𝑖=𝑛𝑗+1	𝑖=𝑛𝑗+1	PUNCT
cana-1747	191	1	|	|	ADV
cana-1747	191	2	√2∑	√2∑	VERB
cana-1747	191	3	𝑏𝑖	𝑏𝑖	ADP
cana-1747	191	4	2	2	NUM
cana-1747	191	5	ln	ln	NOUN
cana-1747	191	6	ln	ln	ADJ
cana-1747	191	7	(	(	PUNCT
cana-1747	191	8	1	1	NUM
cana-1747	191	9	√∑	√∑	NOUN
cana-1747	191	10	𝑏𝑖	𝑏𝑖	ADP
cana-1747	191	11	2	2	NUM
cana-1747	191	12	∞	∞	NUM
cana-1747	191	13	𝑖=𝑛𝑗+1	𝑖=𝑛𝑗+1	PUNCT
cana-1747	191	14	)	)	PUNCT
cana-1747	192	1	∞	∞	NUM
cana-1747	192	2	𝑖=𝑛𝑗+1	𝑖=𝑛𝑗+1	PUNCT
cana-1747	193	1	∞	∞	NUM
cana-1747	193	2	𝑗=1	𝑗=1	X
cana-1747	193	3	>	>	X
cana-1747	194	1	√	√	INTJ
cana-1747	195	1	(	(	PUNCT
cana-1747	195	2	2	2	NUM
cana-1747	195	3	−	−	NOUN
cana-1747	195	4	𝛼	𝛼	NOUN
cana-1747	195	5	)	)	PUNCT
cana-1747	195	6	(	(	PUNCT
cana-1747	195	7	1	1	NUM
cana-1747	195	8	−	−	NOUN
cana-1747	195	9	𝜖2	𝜖2	NOUN
cana-1747	195	10	−	−	PROPN
cana-1747	195	11	1	1	NUM
cana-1747	195	12	𝜃	𝜃	NOUN
cana-1747	195	13	)	)	PUNCT
cana-1747	195	14	2(1	2(1	NUM
cana-1747	196	1	+	+	PUNCT
cana-1747	196	2	𝛽	𝛽	NOUN
cana-1747	196	3	)	)	PUNCT
cana-1747	196	4	−	−	PROPN
cana-1747	196	5	2√	2√	PROPN
cana-1747	196	6	(	(	PUNCT
cana-1747	196	7	1	1	NUM
cana-1747	196	8	−	−	NOUN
cana-1747	196	9	𝜖2	𝜖2	NOUN
cana-1747	196	10	)	)	PUNCT
cana-1747	196	11	𝜃	𝜃	NOUN
cana-1747	196	12	(	(	PUNCT
cana-1747	196	13	1	1	NUM
cana-1747	196	14	+	+	NUM
cana-1747	196	15	𝛼	𝛼	X
cana-1747	196	16	)	)	PUNCT
cana-1747	196	17	}	}	PUNCT
cana-1747	197	1	|	|	ADV
cana-1747	197	2	|	|	ADV
cana-1747	197	3	|	|	ADV
cana-1747	197	4	≥∑	≥∑	PROPN
cana-1747	197	5	1	1	NUM
cana-1747	197	6	6(𝑗	6(𝑗	NUM
cana-1747	197	7	ln	ln	NOUN
cana-1747	197	8	𝜃	𝜃	NOUN
cana-1747	197	9	)	)	PUNCT
cana-1747	197	10	(	(	PUNCT
cana-1747	197	11	2−𝛼	2−𝛼	NUM
cana-1747	197	12	)	)	PUNCT
cana-1747	197	13	2	2	NUM
cana-1747	197	14	∞	∞	NUM
cana-1747	197	15	𝑗=1	𝑗=1	PUNCT
cana-1747	198	1	=	=	PUNCT
cana-1747	198	2	1	1	NUM
cana-1747	198	3	6(ln	6(ln	NUM
cana-1747	198	4	𝜃	𝜃	NOUN
cana-1747	198	5	)	)	PUNCT
cana-1747	198	6	(	(	PUNCT
cana-1747	198	7	2−𝛼	2−𝛼	NUM
cana-1747	198	8	)	)	PUNCT
cana-1747	198	9	2	2	NUM
cana-1747	198	10	∑	∑	SYM
cana-1747	198	11	1	1	NUM
cana-1747	198	12	𝑗	𝑗	PROPN
cana-1747	198	13	(	(	PUNCT
cana-1747	198	14	2−𝛼	2−𝛼	NUM
cana-1747	198	15	)	)	PUNCT
cana-1747	198	16	2	2	NUM
cana-1747	198	17	=	=	SYM
cana-1747	198	18	∞.	∞.	PROPN
cana-1747	198	19	∞	∞	PROPN
cana-1747	198	20	𝑗=1	𝑗=1	PROPN
cana-1747	199	1	here	here	ADV
cana-1747	199	2	we	we	PRON
cana-1747	199	3	note	note	VERB
cana-1747	199	4	that	that	SCONJ
cana-1747	199	5	{	{	PUNCT
cana-1747	199	6	∑	∑	PUNCT
cana-1747	199	7	𝑏𝑖𝑢𝑖(𝑡	𝑏𝑖𝑢𝑖(𝑡	PROPN
cana-1747	199	8	)	)	PUNCT
cana-1747	199	9	∞	∞	NUM
cana-1747	199	10	𝑖=𝑛𝑗+1	𝑖=𝑛𝑗+1	PUNCT
cana-1747	200	1	−	−	X
cana-1747	200	2	∑	∑	PUNCT
cana-1747	200	3	𝑏𝑖𝑢𝑖(𝑡	𝑏𝑖𝑢𝑖(𝑡	PROPN
cana-1747	200	4	)	)	PUNCT
cana-1747	200	5	∞	∞	NUM
cana-1747	200	6	𝑖=𝑛𝑗+1	𝑖=𝑛𝑗+1	PUNCT
cana-1747	201	1	+	+	ADJ
cana-1747	201	2	1	1	NUM
cana-1747	201	3	}	}	PUNCT
cana-1747	201	4	𝑖=1	𝑖=1	PROPN
cana-1747	201	5	∞	∞	PROPN
cana-1747	201	6	is	be	AUX
cana-1747	201	7	a	a	DET
cana-1747	201	8	sequence	sequence	NOUN
cana-1747	201	9	of	of	ADP
cana-1747	201	10	independent	independent	ADJ
cana-1747	201	11	random	random	ADJ
cana-1747	201	12	variables	variable	NOUN
cana-1747	201	13	.	.	PUNCT
cana-1747	202	1	apply	apply	VERB
cana-1747	202	2	lemma	lemma	PROPN
cana-1747	202	3	5	5	NUM
cana-1747	202	4	for	for	ADP
cana-1747	202	5	a.e	a.e	PROPN
cana-1747	202	6	.	.	PROPN
cana-1747	202	7	𝑡	𝑡	PROPN
cana-1747	202	8	,	,	PUNCT
cana-1747	202	9	there	there	PRON
cana-1747	202	10	exists	exist	VERB
cana-1747	202	11	an	an	DET
cana-1747	202	12	infinite	infinite	ADJ
cana-1747	202	13	sequence	sequence	NOUN
cana-1747	202	14	𝑛1	𝑛1	NOUN
cana-1747	202	15	<	<	X
cana-1747	202	16	𝑛2	𝑛2	NOUN
cana-1747	202	17	<	<	X
cana-1747	202	18	𝑛3	𝑛3	PROPN
cana-1747	202	19	<	<	X
cana-1747	202	20	⋯	⋯	VERB
cana-1747	202	21	such	such	ADJ
cana-1747	202	22	that	that	PRON
cana-1747	202	23	,	,	PUNCT
cana-1747	202	24	|∑	|∑	PROPN
cana-1747	202	25	𝑏𝑖𝑢𝑖(𝑡	𝑏𝑖𝑢𝑖(𝑡	NOUN
cana-1747	202	26	)	)	PUNCT
cana-1747	202	27	∞	∞	NUM
cana-1747	202	28	𝑖=𝑛𝑗+1	𝑖=𝑛𝑗+1	PUNCT
cana-1747	203	1	+	+	NOUN
cana-1747	203	2	1	1	NUM
cana-1747	203	3	−	−	NOUN
cana-1747	203	4	∑	∑	ADV
cana-1747	203	5	𝑏𝑖𝑢𝑖(𝑡	𝑏𝑖𝑢𝑖(𝑡	PROPN
cana-1747	203	6	)	)	PUNCT
cana-1747	203	7	∞	∞	NUM
cana-1747	203	8	𝑖=𝑛𝑗+1	𝑖=𝑛𝑗+1	PUNCT
cana-1747	204	1	|	|	ADV
cana-1747	204	2	√2∑	√2∑	VERB
cana-1747	204	3	𝑏𝑖	𝑏𝑖	ADP
cana-1747	204	4	2	2	NUM
cana-1747	204	5	ln	ln	NOUN
cana-1747	204	6	ln	ln	NOUN
cana-1747	204	7	(	(	PUNCT
cana-1747	204	8	1	1	NUM
cana-1747	204	9	√∑	√∑	NOUN
cana-1747	204	10	𝑏𝑖	𝑏𝑖	ADP
cana-1747	204	11	2	2	NUM
cana-1747	204	12	∞	∞	NUM
cana-1747	204	13	𝑖=𝑛𝑗+1	𝑖=𝑛𝑗+1	PUNCT
cana-1747	204	14	)	)	PUNCT
cana-1747	205	1	∞	∞	NUM
cana-1747	205	2	𝑖=𝑛𝑗+1	𝑖=𝑛𝑗+1	PUNCT
cana-1747	205	3	>	>	X
cana-1747	205	4	√	√	PUNCT
cana-1747	205	5	(	(	PUNCT
cana-1747	205	6	2	2	NUM
cana-1747	205	7	−	−	NOUN
cana-1747	205	8	𝛼	𝛼	NOUN
cana-1747	205	9	)	)	PUNCT
cana-1747	205	10	(	(	PUNCT
cana-1747	205	11	1	1	NUM
cana-1747	205	12	−	−	NOUN
cana-1747	205	13	𝜖2	𝜖2	NOUN
cana-1747	205	14	−	−	PROPN
cana-1747	205	15	1	1	NUM
cana-1747	205	16	𝜃	𝜃	NOUN
cana-1747	205	17	)	)	PUNCT
cana-1747	205	18	2(1	2(1	NUM
cana-1747	206	1	+	+	PUNCT
cana-1747	206	2	𝛽	𝛽	NOUN
cana-1747	206	3	)	)	PUNCT
cana-1747	206	4	−	−	PROPN
cana-1747	206	5	2√	2√	PROPN
cana-1747	206	6	(	(	PUNCT
cana-1747	206	7	1	1	NUM
cana-1747	206	8	−	−	NOUN
cana-1747	206	9	𝜖2	𝜖2	NOUN
cana-1747	206	10	)	)	PUNCT
cana-1747	206	11	𝜃	𝜃	NOUN
cana-1747	206	12	(	(	PUNCT
cana-1747	206	13	1	1	NUM
cana-1747	206	14	+	+	NUM
cana-1747	206	15	𝛼	𝛼	X
cana-1747	206	16	)	)	PUNCT
cana-1747	206	17	by	by	ADP
cana-1747	206	18	triangle	triangle	NOUN
cana-1747	206	19	inequality	inequality	NOUN
cana-1747	206	20	,	,	PUNCT
cana-1747	206	21	we	we	PRON
cana-1747	206	22	have	have	AUX
cana-1747	206	23	|∑	|∑	VERB
cana-1747	206	24	𝑏𝑖𝑢𝑖(𝑡	𝑏𝑖𝑢𝑖(𝑡	NOUN
cana-1747	206	25	)	)	PUNCT
cana-1747	206	26	∞	∞	NUM
cana-1747	206	27	𝑖=𝑛𝑗+1	𝑖=𝑛𝑗+1	PUNCT
cana-1747	207	1	+	+	ADJ
cana-1747	207	2	1	1	NUM
cana-1747	207	3	|	|	NOUN
cana-1747	207	4	√2∑	√2∑	VERB
cana-1747	207	5	𝑏𝑖	𝑏𝑖	ADP
cana-1747	207	6	2	2	NUM
cana-1747	207	7	ln	ln	NOUN
cana-1747	207	8	ln	ln	NOUN
cana-1747	207	9	(	(	PUNCT
cana-1747	207	10	1	1	NUM
cana-1747	207	11	√∑	√∑	NOUN
cana-1747	207	12	𝑏𝑖	𝑏𝑖	ADP
cana-1747	207	13	2	2	NUM
cana-1747	207	14	∞	∞	NUM
cana-1747	207	15	𝑖=𝑛𝑗+1	𝑖=𝑛𝑗+1	PUNCT
cana-1747	207	16	)	)	PUNCT
cana-1747	208	1	∞	∞	NUM
cana-1747	208	2	𝑖=𝑛𝑗+1	𝑖=𝑛𝑗+1	PUNCT
cana-1747	209	1	+	+	CCONJ
cana-1747	209	2	|∑	|∑	NUM
cana-1747	209	3	𝑏𝑖𝑢𝑖(𝑡	𝑏𝑖𝑢𝑖(𝑡	NOUN
cana-1747	209	4	)	)	PUNCT
cana-1747	209	5	∞	∞	NUM
cana-1747	209	6	𝑖=𝑛𝑗+1	𝑖=𝑛𝑗+1	PUNCT
cana-1747	210	1	|	|	ADV
cana-1747	210	2	√2∑	√2∑	VERB
cana-1747	210	3	𝑏𝑖	𝑏𝑖	ADP
cana-1747	210	4	2	2	NUM
cana-1747	210	5	ln	ln	NOUN
cana-1747	210	6	ln	ln	ADJ
cana-1747	210	7	(	(	PUNCT
cana-1747	210	8	1	1	NUM
cana-1747	210	9	√∑	√∑	NOUN
cana-1747	210	10	𝑏𝑖	𝑏𝑖	ADP
cana-1747	210	11	2	2	NUM
cana-1747	210	12	∞	∞	NUM
cana-1747	210	13	𝑖=𝑛𝑗+1	𝑖=𝑛𝑗+1	PUNCT
cana-1747	210	14	)	)	PUNCT
cana-1747	211	1	∞	∞	NUM
cana-1747	211	2	𝑖=𝑛𝑗+1	𝑖=𝑛𝑗+1	PUNCT
cana-1747	212	1	communications	communication	NOUN
cana-1747	212	2	on	on	ADP
cana-1747	212	3	applied	apply	VERB
cana-1747	212	4	nonlinear	nonlinear	ADJ
cana-1747	212	5	analysis	analysis	NOUN
cana-1747	212	6	issn	issn	NOUN
cana-1747	212	7	:	:	PUNCT
cana-1747	212	8	1074	1074	NUM
cana-1747	212	9	-	-	PUNCT
cana-1747	212	10	133x	133x	NUM
cana-1747	212	11	vol	vol	NOUN
cana-1747	212	12	32	32	NUM
cana-1747	212	13	no	no	NOUN
cana-1747	212	14	.	.	NOUN
cana-1747	212	15	2	2	NUM
cana-1747	212	16	(	(	PUNCT
cana-1747	212	17	2025	2025	NUM
cana-1747	212	18	)	)	PUNCT
cana-1747	212	19	364	364	NUM
cana-1747	212	20	https://internationalpubls.com	https://internationalpubls.com	X
cana-1747	212	21	(	(	PUNCT
cana-1747	212	22	3.4	3.4	NUM
cana-1747	212	23	)	)	PUNCT
cana-1747	212	24	>	>	X
cana-1747	212	25	√	√	INTJ
cana-1747	212	26	(	(	PUNCT
cana-1747	212	27	2	2	NUM
cana-1747	212	28	−	−	NOUN
cana-1747	212	29	𝛼	𝛼	NOUN
cana-1747	212	30	)	)	PUNCT
cana-1747	212	31	(	(	PUNCT
cana-1747	212	32	1	1	NUM
cana-1747	212	33	−	−	NOUN
cana-1747	212	34	𝜖2	𝜖2	NOUN
cana-1747	212	35	−	−	PROPN
cana-1747	212	36	1	1	NUM
cana-1747	212	37	𝜃	𝜃	NOUN
cana-1747	212	38	)	)	PUNCT
cana-1747	212	39	2(1	2(1	NUM
cana-1747	213	1	+	+	PUNCT
cana-1747	213	2	𝛽	𝛽	NOUN
cana-1747	213	3	)	)	PUNCT
cana-1747	213	4	−	−	PROPN
cana-1747	213	5	2√	2√	PROPN
cana-1747	213	6	(	(	PUNCT
cana-1747	213	7	1	1	NUM
cana-1747	213	8	−	−	NOUN
cana-1747	213	9	𝜖2	𝜖2	NOUN
cana-1747	213	10	)	)	PUNCT
cana-1747	213	11	𝜃	𝜃	NOUN
cana-1747	213	12	(	(	PUNCT
cana-1747	213	13	1	1	NUM
cana-1747	213	14	+	+	NUM
cana-1747	213	15	𝛼	𝛼	X
cana-1747	213	16	)	)	PUNCT
cana-1747	213	17	.	.	PUNCT
cana-1747	214	1	we	we	PRON
cana-1747	214	2	have	have	VERB
cana-1747	214	3	|𝐴|	|𝐴|	NOUN
cana-1747	214	4	=	=	SYM
cana-1747	214	5	|{𝑡	|{𝑡	NOUN
cana-1747	214	6	∈	∈	NOUN
cana-1747	214	7	𝐼	𝐼	NOUN
cana-1747	214	8	:	:	PUNCT
cana-1747	214	9	sup	sup	NOUN
cana-1747	214	10	n≥nj+1	n≥nj+1	NOUN
cana-1747	214	11	|∑	|∑	VERB
cana-1747	215	1	𝑏𝑖𝑢𝑖(𝑡	𝑏𝑖𝑢𝑖(𝑡	NOUN
cana-1747	215	2	)	)	PUNCT
cana-1747	216	1	∞	∞	NUM
cana-1747	216	2	𝑖=𝑛+1	𝑖=𝑛+1	NOUN
cana-1747	217	1	|	|	ADV
cana-1747	217	2	>	>	X
cana-1747	218	1	√	√	NUM
cana-1747	218	2	2(1	2(1	NUM
cana-1747	219	1	+	+	PUNCT
cana-1747	219	2	𝛼	𝛼	X
cana-1747	219	3	)	)	PUNCT
cana-1747	219	4	𝜃	𝜃	NOUN
cana-1747	219	5	∑	∑	PROPN
cana-1747	219	6	𝑏𝑖	𝑏𝑖	ADP
cana-1747	219	7	2	2	NUM
cana-1747	219	8	(	(	PUNCT
cana-1747	219	9	1	1	NUM
cana-1747	219	10	−	−	NOUN
cana-1747	219	11	𝜖2	𝜖2	NOUN
cana-1747	219	12	)	)	PUNCT
cana-1747	219	13	∞	∞	NUM
cana-1747	219	14	𝑖=𝑛𝑗+1	𝑖=𝑛𝑗+1	PUNCT
cana-1747	220	1	ln	ln	ADV
cana-1747	220	2	ln	ln	ADJ
cana-1747	220	3	(	(	PUNCT
cana-1747	220	4	1	1	NUM
cana-1747	220	5	∑	∑	ADV
cana-1747	220	6	𝑏𝑖	𝑏𝑖	ADP
cana-1747	220	7	2	2	NUM
cana-1747	220	8	∞	∞	NUM
cana-1747	220	9	𝑖=𝑛𝑗+1	𝑖=𝑛𝑗+1	PUNCT
cana-1747	220	10	)	)	PUNCT
cana-1747	220	11	}	}	PUNCT
cana-1747	220	12	|	|	CCONJ
cana-1747	220	13	<	<	X
cana-1747	220	14	24	24	NUM
cana-1747	220	15	(	(	PUNCT
cana-1747	220	16	ln	ln	ADJ
cana-1747	220	17	𝜃)(1+𝛼)(1−𝜖	𝜃)(1+𝛼)(1−𝜖	NOUN
cana-1747	220	18	2	2	NUM
cana-1747	220	19	)	)	PUNCT
cana-1747	220	20	1	1	NUM
cana-1747	220	21	𝑗(1+𝛼)(1−𝜖	𝑗(1+𝛼)(1−𝜖	NOUN
cana-1747	220	22	2	2	NUM
cana-1747	220	23	)	)	PUNCT
cana-1747	220	24	.	.	PUNCT
cana-1747	221	1	so	so	ADV
cana-1747	221	2	∑	∑	PUNCT
cana-1747	221	3	|{𝑡	|{𝑡	X
cana-1747	221	4	∈	∈	NOUN
cana-1747	221	5	𝐼	𝐼	PROPN
cana-1747	221	6	:	:	PUNCT
cana-1747	221	7	sup	sup	NOUN
cana-1747	221	8	n≥nj+1	n≥nj+1	NOUN
cana-1747	221	9	|∑	|∑	VERB
cana-1747	221	10	𝑏𝑖𝑢𝑖(𝑡	𝑏𝑖𝑢𝑖(𝑡	NOUN
cana-1747	221	11	)	)	PUNCT
cana-1747	221	12	∞	∞	NUM
cana-1747	221	13	𝑖=𝑛+1	𝑖=𝑛+1	NOUN
cana-1747	222	1	|	|	ADV
cana-1747	222	2	>	>	X
cana-1747	222	3	√	√	NUM
cana-1747	222	4	2(1+𝛼	2(1+𝛼	NUM
cana-1747	222	5	)	)	PUNCT
cana-1747	222	6	𝜃	𝜃	NOUN
cana-1747	222	7	∑	∑	NOUN
cana-1747	222	8	𝑏𝑖	𝑏𝑖	ADP
cana-1747	222	9	2(1	2(1	NUM
cana-1747	222	10	−	−	NOUN
cana-1747	222	11	𝜖2	𝜖2	NOUN
cana-1747	222	12	)	)	PUNCT
cana-1747	222	13	∞	∞	NUM
cana-1747	222	14	𝑖=𝑛𝑗+1	𝑖=𝑛𝑗+1	PUNCT
cana-1747	223	1	ln	ln	ADV
cana-1747	223	2	ln	ln	ADJ
cana-1747	223	3	(	(	PUNCT
cana-1747	223	4	1	1	NUM
cana-1747	223	5	∑	∑	ADV
cana-1747	223	6	𝑏𝑖	𝑏𝑖	ADP
cana-1747	223	7	2	2	NUM
cana-1747	223	8	∞	∞	NUM
cana-1747	223	9	𝑖=𝑛𝑗+1	𝑖=𝑛𝑗+1	PUNCT
cana-1747	223	10	)	)	PUNCT
cana-1747	223	11	}	}	PUNCT
cana-1747	223	12	|∞	|∞	NOUN
cana-1747	224	1	𝑗=1	𝑗=1	PUNCT
cana-1747	224	2	<	<	X
cana-1747	224	3	∑	∑	PROPN
cana-1747	224	4	24	24	NUM
cana-1747	224	5	(	(	PUNCT
cana-1747	224	6	ln	ln	ADJ
cana-1747	224	7	𝜃)(1+𝛼)(1−𝜖	𝜃)(1+𝛼)(1−𝜖	NOUN
cana-1747	224	8	2	2	NUM
cana-1747	224	9	)	)	PUNCT
cana-1747	224	10	1	1	NUM
cana-1747	224	11	𝑗(1+𝛼)(1−𝜖	𝑗(1+𝛼)(1−𝜖	NOUN
cana-1747	224	12	2	2	NUM
cana-1747	224	13	)	)	PUNCT
cana-1747	224	14	∞	∞	NUM
cana-1747	224	15	𝑗=1	𝑗=1	PUNCT
cana-1747	225	1	=	=	SYM
cana-1747	225	2	24	24	NUM
cana-1747	225	3	(	(	PUNCT
cana-1747	225	4	ln	ln	ADJ
cana-1747	225	5	𝜃)(1+𝛼)(1−𝜖	𝜃)(1+𝛼)(1−𝜖	NOUN
cana-1747	225	6	2	2	NUM
cana-1747	225	7	)	)	PUNCT
cana-1747	225	8	∑	∑	ADP
cana-1747	225	9	1	1	NUM
cana-1747	225	10	𝑗(1+𝛼)(1−𝜖	𝑗(1+𝛼)(1−𝜖	NOUN
cana-1747	225	11	2	2	NUM
cana-1747	225	12	)	)	PUNCT
cana-1747	225	13	∞	∞	NUM
cana-1747	225	14	𝑗=1	𝑗=1	X
cana-1747	225	15	<	<	X
cana-1747	225	16	∞.	∞.	PROPN
cana-1747	225	17	applying	apply	VERB
cana-1747	225	18	lemma	lemma	PROPN
cana-1747	225	19	6	6	NUM
cana-1747	225	20	,	,	PUNCT
cana-1747	225	21	for	for	ADP
cana-1747	225	22	a.e	a.e	PROPN
cana-1747	225	23	.	.	PROPN
cana-1747	225	24	𝑡	𝑡	PROPN
cana-1747	225	25	,	,	PUNCT
cana-1747	225	26	we	we	PRON
cana-1747	225	27	get	get	VERB
cana-1747	225	28	sup	sup	NOUN
cana-1747	225	29	n≥nj+1	n≥nj+1	NOUN
cana-1747	225	30	|∑	|∑	VERB
cana-1747	226	1	𝑏𝑖𝑢𝑖(𝑡	𝑏𝑖𝑢𝑖(𝑡	NOUN
cana-1747	226	2	)	)	PUNCT
cana-1747	226	3	∞	∞	NUM
cana-1747	226	4	𝑖=𝑛+1	𝑖=𝑛+1	NOUN
cana-1747	227	1	|	|	ADV
cana-1747	227	2	≤	≤	NUM
cana-1747	227	3	√	√	CCONJ
cana-1747	227	4	2(1	2(1	NUM
cana-1747	228	1	+	+	PUNCT
cana-1747	228	2	𝛼	𝛼	X
cana-1747	228	3	)	)	PUNCT
cana-1747	228	4	𝜃	𝜃	NOUN
cana-1747	228	5	∑	∑	PROPN
cana-1747	228	6	𝑏𝑖	𝑏𝑖	ADP
cana-1747	228	7	2(1	2(1	NUM
cana-1747	228	8	−	−	NOUN
cana-1747	228	9	𝜖2	𝜖2	NOUN
cana-1747	228	10	)	)	PUNCT
cana-1747	228	11	∞	∞	NUM
cana-1747	228	12	𝑖=𝑛𝑗+1	𝑖=𝑛𝑗+1	PUNCT
cana-1747	229	1	ln	ln	ADV
cana-1747	229	2	ln	ln	ADJ
cana-1747	229	3	(	(	PUNCT
cana-1747	229	4	1	1	NUM
cana-1747	229	5	∑	∑	ADV
cana-1747	229	6	𝑏𝑖	𝑏𝑖	ADP
cana-1747	229	7	2	2	NUM
cana-1747	229	8	∞	∞	NUM
cana-1747	229	9	𝑖=𝑛𝑗+1	𝑖=𝑛𝑗+1	PUNCT
cana-1747	229	10	)	)	PUNCT
cana-1747	229	11	i.e.	i.e.	X
cana-1747	229	12	(	(	PUNCT
cana-1747	229	13	3.5	3.5	NUM
cana-1747	229	14	)	)	PUNCT
cana-1747	229	15	sup	sup	NOUN
cana-1747	229	16	n≥nj+1	n≥nj+1	NOUN
cana-1747	229	17	|∑	|∑	VERB
cana-1747	229	18	𝑏𝑖𝑢𝑖(𝑡	𝑏𝑖𝑢𝑖(𝑡	NOUN
cana-1747	229	19	)	)	PUNCT
cana-1747	229	20	∞	∞	NUM
cana-1747	229	21	𝑖=𝑛+1	𝑖=𝑛+1	NOUN
cana-1747	230	1	|	|	ADV
cana-1747	230	2	√∑	√∑	VERB
cana-1747	230	3	𝑏𝑖	𝑏𝑖	ADP
cana-1747	230	4	2	2	NUM
cana-1747	230	5	∞	∞	NUM
cana-1747	230	6	𝑖=𝑛𝑗+1	𝑖=𝑛𝑗+1	PUNCT
cana-1747	231	1	2	2	NUM
cana-1747	231	2	ln	ln	NOUN
cana-1747	231	3	ln	ln	ADJ
cana-1747	231	4	(	(	PUNCT
cana-1747	231	5	1	1	NUM
cana-1747	231	6	∑	∑	ADV
cana-1747	231	7	𝑏𝑖	𝑏𝑖	ADP
cana-1747	231	8	2	2	NUM
cana-1747	231	9	∞	∞	NUM
cana-1747	231	10	𝑖=𝑛𝑗+1	𝑖=𝑛𝑗+1	PUNCT
cana-1747	231	11	)	)	PUNCT
cana-1747	231	12	≤	≤	NUM
cana-1747	231	13	√	√	NUM
cana-1747	231	14	(	(	PUNCT
cana-1747	231	15	1	1	NUM
cana-1747	231	16	−	−	NOUN
cana-1747	231	17	𝜖2)(1	𝜖2)(1	PROPN
cana-1747	231	18	+	+	NUM
cana-1747	231	19	𝛼	𝛼	X
cana-1747	231	20	)	)	PUNCT
cana-1747	231	21	𝜃	𝜃	NOUN
cana-1747	231	22	for	for	ADP
cana-1747	231	23	sufficiently	sufficiently	ADV
cana-1747	231	24	large	large	ADJ
cana-1747	231	25	𝑗	𝑗	PRON
cana-1747	231	26	≥	≥	NOUN
cana-1747	231	27	𝑁	𝑁	PROPN
cana-1747	231	28	(	(	PUNCT
cana-1747	231	29	say	say	INTJ
cana-1747	231	30	)	)	PUNCT
cana-1747	231	31	.	.	PUNCT
cana-1747	232	1	thus	thus	ADV
cana-1747	232	2	from	from	ADP
cana-1747	232	3	(	(	PUNCT
cana-1747	232	4	3.4	3.4	NUM
cana-1747	232	5	)	)	PUNCT
cana-1747	232	6	and	and	CCONJ
cana-1747	232	7	(	(	PUNCT
cana-1747	232	8	3.5	3.5	NUM
cana-1747	232	9	)	)	PUNCT
cana-1747	232	10	,	,	PUNCT
cana-1747	232	11	for	for	ADP
cana-1747	232	12	a.e	a.e	PROPN
cana-1747	232	13	.	.	PROPN
cana-1747	232	14	𝑡	𝑡	NOUN
cana-1747	232	15	we	we	PRON
cana-1747	232	16	get	get	VERB
cana-1747	232	17	𝑛1	𝑛1	ADJ
cana-1747	232	18	<	<	X
cana-1747	232	19	𝑛2	𝑛2	NOUN
cana-1747	232	20	<	<	X
cana-1747	232	21	𝑛3	𝑛3	PROPN
cana-1747	232	22	<	<	X
cana-1747	232	23	⋯	⋯	VERB
cana-1747	232	24	such	such	ADJ
cana-1747	232	25	that	that	PRON
cana-1747	232	26	,	,	PUNCT
cana-1747	232	27	|∑	|∑	PROPN
cana-1747	232	28	𝑏𝑖𝑢𝑖(𝑡	𝑏𝑖𝑢𝑖(𝑡	NOUN
cana-1747	232	29	)	)	PUNCT
cana-1747	232	30	∞	∞	NUM
cana-1747	232	31	𝑖=𝑛𝑗+1	𝑖=𝑛𝑗+1	PUNCT
cana-1747	233	1	|	|	ADV
cana-1747	233	2	√2∑	√2∑	VERB
cana-1747	233	3	𝑏𝑖	𝑏𝑖	ADP
cana-1747	233	4	2	2	NUM
cana-1747	233	5	ln	ln	NOUN
cana-1747	233	6	ln	ln	ADJ
cana-1747	233	7	(	(	PUNCT
cana-1747	233	8	1	1	NUM
cana-1747	233	9	√∑	√∑	NOUN
cana-1747	233	10	𝑏𝑖	𝑏𝑖	ADP
cana-1747	233	11	2	2	NUM
cana-1747	233	12	∞	∞	NUM
cana-1747	233	13	𝑖=𝑛𝑗+1	𝑖=𝑛𝑗+1	PUNCT
cana-1747	233	14	)	)	PUNCT
cana-1747	234	1	∞	∞	NUM
cana-1747	234	2	𝑖=𝑛𝑗+1	𝑖=𝑛𝑗+1	PUNCT
cana-1747	234	3	>	>	X
cana-1747	234	4	√	√	PUNCT
cana-1747	234	5	(	(	PUNCT
cana-1747	234	6	2	2	NUM
cana-1747	234	7	−	−	NOUN
cana-1747	234	8	𝛼	𝛼	NOUN
cana-1747	234	9	)	)	PUNCT
cana-1747	234	10	(	(	PUNCT
cana-1747	234	11	1	1	NUM
cana-1747	234	12	−	−	NOUN
cana-1747	234	13	𝜖2	𝜖2	NOUN
cana-1747	234	14	−	−	PROPN
cana-1747	234	15	1	1	NUM
cana-1747	234	16	𝜃	𝜃	NOUN
cana-1747	234	17	)	)	PUNCT
cana-1747	234	18	2(1	2(1	NUM
cana-1747	235	1	+	+	PUNCT
cana-1747	235	2	𝛽	𝛽	NOUN
cana-1747	235	3	)	)	PUNCT
cana-1747	235	4	−	−	PROPN
cana-1747	235	5	3√	3√	NOUN
cana-1747	235	6	(	(	PUNCT
cana-1747	235	7	1	1	NUM
cana-1747	235	8	−	−	NOUN
cana-1747	235	9	𝜖2	𝜖2	NOUN
cana-1747	235	10	)	)	PUNCT
cana-1747	235	11	𝜃	𝜃	NOUN
cana-1747	235	12	(	(	PUNCT
cana-1747	235	13	1	1	NUM
cana-1747	235	14	+	+	NUM
cana-1747	235	15	𝛼	𝛼	X
cana-1747	235	16	)	)	PUNCT
cana-1747	235	17	.	.	PUNCT
cana-1747	236	1	consequently	consequently	ADV
cana-1747	236	2	,	,	PUNCT
cana-1747	236	3	for	for	ADP
cana-1747	236	4	a.e	a.e	PROPN
cana-1747	236	5	.	.	PROPN
cana-1747	236	6	𝑡	𝑡	PROPN
cana-1747	236	7	communications	communication	NOUN
cana-1747	236	8	on	on	ADP
cana-1747	236	9	applied	apply	VERB
cana-1747	236	10	nonlinear	nonlinear	ADJ
cana-1747	236	11	analysis	analysis	NOUN
cana-1747	236	12	issn	issn	NOUN
cana-1747	236	13	:	:	PUNCT
cana-1747	236	14	1074	1074	NUM
cana-1747	236	15	-	-	PUNCT
cana-1747	236	16	133x	133x	NUM
cana-1747	236	17	vol	vol	NOUN
cana-1747	236	18	32	32	NUM
cana-1747	236	19	no	no	NOUN
cana-1747	236	20	.	.	NOUN
cana-1747	236	21	2	2	NUM
cana-1747	236	22	(	(	PUNCT
cana-1747	236	23	2025	2025	NUM
cana-1747	236	24	)	)	PUNCT
cana-1747	237	1	365	365	NUM
cana-1747	237	2	https://internationalpubls.com	https://internationalpubls.com	X
cana-1747	237	3	|∑	|∑	ADV
cana-1747	237	4	𝑏𝑖𝑢𝑖(𝑡	𝑏𝑖𝑢𝑖(𝑡	NOUN
cana-1747	237	5	)	)	PUNCT
cana-1747	237	6	∞	∞	PROPN
cana-1747	237	7	𝑖=𝑛+1	𝑖=𝑛+1	NOUN
cana-1747	237	8	|	|	ADV
cana-1747	237	9	√2∑	√2∑	VERB
cana-1747	237	10	𝑏𝑖	𝑏𝑖	ADP
cana-1747	237	11	2	2	NUM
cana-1747	237	12	ln	ln	NOUN
cana-1747	237	13	ln	ln	NOUN
cana-1747	237	14	(	(	PUNCT
cana-1747	237	15	1	1	NUM
cana-1747	237	16	√∑	√∑	NOUN
cana-1747	237	17	𝑏𝑖	𝑏𝑖	ADP
cana-1747	237	18	2	2	NUM
cana-1747	237	19	∞	∞	NUM
cana-1747	237	20	𝑖=𝑛+1	𝑖=𝑛+1	NOUN
cana-1747	237	21	)	)	PUNCT
cana-1747	238	1	∞	∞	NUM
cana-1747	238	2	𝑖=𝑛+1	𝑖=𝑛+1	NOUN
cana-1747	238	3	>	>	X
cana-1747	238	4	√	√	INTJ
cana-1747	238	5	(	(	PUNCT
cana-1747	238	6	2	2	NUM
cana-1747	238	7	−	−	NOUN
cana-1747	238	8	𝛼	𝛼	NOUN
cana-1747	238	9	)	)	PUNCT
cana-1747	238	10	(	(	PUNCT
cana-1747	238	11	1	1	NUM
cana-1747	238	12	−	−	NOUN
cana-1747	238	13	𝜖2	𝜖2	NOUN
cana-1747	238	14	−	−	PROPN
cana-1747	238	15	1	1	NUM
cana-1747	238	16	𝜃	𝜃	NOUN
cana-1747	238	17	)	)	PUNCT
cana-1747	238	18	2(1	2(1	NUM
cana-1747	239	1	+	+	PUNCT
cana-1747	239	2	𝛽	𝛽	NOUN
cana-1747	239	3	)	)	PUNCT
cana-1747	239	4	−	−	PROPN
cana-1747	239	5	3√	3√	NOUN
cana-1747	239	6	(	(	PUNCT
cana-1747	239	7	1	1	NUM
cana-1747	239	8	−	−	NOUN
cana-1747	239	9	𝜖2	𝜖2	NOUN
cana-1747	239	10	)	)	PUNCT
cana-1747	239	11	𝜃	𝜃	NOUN
cana-1747	239	12	(	(	PUNCT
cana-1747	239	13	1	1	NUM
cana-1747	239	14	+	+	NUM
cana-1747	239	15	𝛼	𝛼	X
cana-1747	239	16	)	)	PUNCT
cana-1747	239	17	.	.	PUNCT
cana-1747	240	1	letting	let	VERB
cana-1747	240	2	𝜃	𝜃	X
cana-1747	240	3	↗	↗	NUM
cana-1747	240	4	∞	∞	PROPN
cana-1747	240	5	,	,	PUNCT
cana-1747	240	6	𝜖	𝜖	PROPN
cana-1747	240	7	,	,	PUNCT
cana-1747	240	8	𝛼	𝛼	PROPN
cana-1747	240	9	,	,	PUNCT
cana-1747	240	10	𝛽	𝛽	PROPN
cana-1747	240	11	↘	↘	PROPN
cana-1747	240	12	0	0	NUM
cana-1747	240	13	,	,	PUNCT
cana-1747	240	14	we	we	PRON
cana-1747	240	15	get	get	VERB
cana-1747	240	16	|∑	|∑	VERB
cana-1747	240	17	𝑏𝑖𝑢𝑖(𝑡	𝑏𝑖𝑢𝑖(𝑡	NOUN
cana-1747	240	18	)	)	PUNCT
cana-1747	240	19	∞	∞	PROPN
cana-1747	240	20	𝑖=𝑛+1	𝑖=𝑛+1	NOUN
cana-1747	241	1	|	|	ADV
cana-1747	241	2	√2∑	√2∑	VERB
cana-1747	241	3	𝑏𝑖	𝑏𝑖	ADP
cana-1747	241	4	2	2	NUM
cana-1747	241	5	ln	ln	NOUN
cana-1747	241	6	ln	ln	NOUN
cana-1747	241	7	(	(	PUNCT
cana-1747	241	8	1	1	NUM
cana-1747	241	9	√∑	√∑	NOUN
cana-1747	241	10	𝑏𝑖	𝑏𝑖	ADP
cana-1747	241	11	2	2	NUM
cana-1747	241	12	∞	∞	NUM
cana-1747	241	13	𝑖=𝑛+1	𝑖=𝑛+1	NOUN
cana-1747	241	14	)	)	PUNCT
cana-1747	242	1	∞	∞	NUM
cana-1747	242	2	𝑖=𝑛+1	𝑖=𝑛+1	NOUN
cana-1747	242	3	≥	≥	NOUN
cana-1747	242	4	1	1	NUM
cana-1747	242	5	.	.	PUNCT
cana-1747	243	1	thus	thus	ADV
cana-1747	243	2	,	,	PUNCT
cana-1747	243	3	limsup	limsup	X
cana-1747	243	4	n→∞	n→∞	X
cana-1747	243	5	|∑	|∑	VERB
cana-1747	243	6	𝑏𝑖𝑢𝑖(𝑡	𝑏𝑖𝑢𝑖(𝑡	NOUN
cana-1747	243	7	)	)	PUNCT
cana-1747	243	8	∞	∞	PROPN
cana-1747	243	9	𝑖=𝑛+1	𝑖=𝑛+1	NOUN
cana-1747	244	1	|	|	ADV
cana-1747	244	2	√2∑	√2∑	VERB
cana-1747	244	3	𝑏𝑖	𝑏𝑖	ADP
cana-1747	244	4	2	2	NUM
cana-1747	244	5	ln	ln	NOUN
cana-1747	244	6	ln	ln	NOUN
cana-1747	244	7	(	(	PUNCT
cana-1747	244	8	1	1	NUM
cana-1747	244	9	√∑	√∑	NOUN
cana-1747	244	10	𝑏𝑖	𝑏𝑖	ADP
cana-1747	244	11	2	2	NUM
cana-1747	244	12	∞	∞	NUM
cana-1747	244	13	𝑖=𝑛+1	𝑖=𝑛+1	NOUN
cana-1747	244	14	)	)	PUNCT
cana-1747	245	1	∞	∞	NUM
cana-1747	245	2	𝑖=𝑛+1	𝑖=𝑛+1	NOUN
cana-1747	245	3	≥	≥	NOUN
cana-1747	245	4	1	1	NUM
cana-1747	245	5	.	.	PUNCT
cana-1747	246	1	this	this	PRON
cana-1747	246	2	gives	give	VERB
cana-1747	246	3	limsup	limsup	PROPN
cana-1747	246	4	n→∞	n→∞	X
cana-1747	246	5	|∑	|∑	VERB
cana-1747	246	6	𝑏𝑖𝑢𝑖(𝑡	𝑏𝑖𝑢𝑖(𝑡	NOUN
cana-1747	246	7	)	)	PUNCT
cana-1747	246	8	∞	∞	NUM
cana-1747	246	9	𝑖=𝑛+1	𝑖=𝑛+1	NOUN
cana-1747	247	1	|	|	ADV
cana-1747	247	2	√2	√2	PROPN
cana-1747	247	3	𝐵𝑛	𝐵𝑛	PROPN
cana-1747	248	1	ln	ln	PROPN
cana-1747	249	1	ln	ln	ADJ
cana-1747	250	1	1	1	NUM
cana-1747	250	2	𝐵𝑛	𝐵𝑛	PROPN
cana-1747	250	3	≥	≥	NUM
cana-1747	250	4	1	1	NUM
cana-1747	250	5	for	for	ADP
cana-1747	250	6	a.e	a.e	PROPN
cana-1747	250	7	.	.	PUNCT
cana-1747	250	8	𝑡	𝑡	PROPN
cana-1747	250	9	∈	∈	PROPN
cana-1747	250	10	(	(	PUNCT
cana-1747	250	11	0	0	NUM
cana-1747	250	12	,	,	PUNCT
cana-1747	250	13	1	1	NUM
cana-1747	250	14	)	)	PUNCT
cana-1747	250	15	.	.	PUNCT
cana-1747	251	1	this	this	PRON
cana-1747	251	2	completes	complete	VERB
cana-1747	251	3	the	the	DET
cana-1747	251	4	proof	proof	NOUN
cana-1747	251	5	of	of	ADP
cana-1747	251	6	the	the	DET
cana-1747	251	7	main	main	ADJ
cana-1747	251	8	theorem	theorem	NOUN
cana-1747	251	9	.	.	PUNCT
cana-1747	252	1	conclusion	conclusion	NOUN
cana-1747	252	2	when	when	SCONJ
cana-1747	252	3	we	we	PRON
cana-1747	252	4	combine	combine	VERB
cana-1747	252	5	the	the	DET
cana-1747	252	6	result	result	NOUN
cana-1747	252	7	of	of	ADP
cana-1747	252	8	theorem	theorem	NOUN
cana-1747	252	9	3	3	NUM
cana-1747	252	10	with	with	ADP
cana-1747	252	11	result	result	NOUN
cana-1747	252	12	obtained	obtain	VERB
cana-1747	252	13	in	in	ADP
cana-1747	252	14	theorem	theorem	NOUN
cana-1747	252	15	4	4	NUM
cana-1747	252	16	,	,	PUNCT
cana-1747	252	17	we	we	PRON
cana-1747	252	18	have	have	AUX
cana-1747	252	19	limsup	limsup	ADJ
cana-1747	252	20	n→∞	n→∞	X
cana-1747	252	21	|∑	|∑	VERB
cana-1747	252	22	𝑏𝑖𝑢𝑖(𝑡	𝑏𝑖𝑢𝑖(𝑡	NOUN
cana-1747	252	23	)	)	PUNCT
cana-1747	253	1	∞	∞	NUM
cana-1747	253	2	𝑖=𝑛+1	𝑖=𝑛+1	NOUN
cana-1747	254	1	|	|	ADV
cana-1747	254	2	√2	√2	PROPN
cana-1747	254	3	𝐵𝑛	𝐵𝑛	PROPN
cana-1747	254	4	ln	ln	PROPN
cana-1747	254	5	ln	ln	ADJ
cana-1747	254	6	1	1	NUM
cana-1747	254	7	𝐵𝑛	𝐵𝑛	PROPN
cana-1747	254	8	=	=	NOUN
cana-1747	254	9	1	1	NUM
cana-1747	254	10	for	for	ADP
cana-1747	254	11	a.	a.	PROPN
cana-1747	254	12	e.	e.	PROPN
cana-1747	254	13	𝑡	𝑡	PROPN
cana-1747	254	14	∈	∈	PROPN
cana-1747	254	15	(	(	PUNCT
cana-1747	254	16	0	0	NUM
cana-1747	254	17	,	,	PUNCT
cana-1747	254	18	1	1	NUM
cana-1747	254	19	)	)	PUNCT
cana-1747	254	20	.	.	PUNCT
cana-1747	255	1	this	this	PRON
cana-1747	255	2	completes	complete	VERB
cana-1747	255	3	the	the	DET
cana-1747	255	4	law	law	NOUN
cana-1747	255	5	of	of	ADP
cana-1747	255	6	the	the	DET
cana-1747	255	7	iterated	iterated	ADJ
cana-1747	255	8	logarithm	logarithm	NOUN
cana-1747	255	9	for	for	ADP
cana-1747	255	10	the	the	DET
cana-1747	255	11	summation	summation	NOUN
cana-1747	255	12	of	of	ADP
cana-1747	255	13	the	the	DET
cana-1747	255	14	signum	signum	PROPN
cana-1747	255	15	functions	function	NOUN
cana-1747	255	16	.	.	PUNCT
cana-1747	256	1	references	reference	NOUN
cana-1747	256	2	[	[	X
cana-1747	256	3	1	1	NUM
cana-1747	256	4	]	]	PUNCT
cana-1747	256	5	a.	a.	NOUN
cana-1747	256	6	khintchine	khintchine	PROPN
cana-1747	256	7	,	,	PUNCT
cana-1747	256	8	uber	uber	ADJ
cana-1747	256	9	einen	einen	PROPN
cana-1747	256	10	sat	sit	VERB
cana-1747	256	11	:	:	PUNCT
cana-1747	256	12	der	der	ADJ
cana-1747	256	13	wahrscheinlichkeitsrechnung	wahrscheinlichkeitsrechnung	NOUN
cana-1747	256	14	,	,	PUNCT
cana-1747	256	15	fundamenta	fundamenta	PROPN
cana-1747	256	16	mathematica	mathematica	PROPN
cana-1747	256	17	,	,	PUNCT
cana-1747	256	18	6(1924	6(1924	NUM
cana-1747	256	19	)	)	PUNCT
cana-1747	256	20	,	,	PUNCT
cana-1747	256	21	9	9	NUM
cana-1747	256	22	-	-	SYM
cana-1747	256	23	20	20	NUM
cana-1747	256	24	.	.	PUNCT
cana-1747	257	1	[	[	X
cana-1747	257	2	2	2	NUM
cana-1747	257	3	]	]	PUNCT
cana-1747	257	4	a.	a.	NOUN
cana-1747	257	5	zygmund	zygmund	PROPN
cana-1747	257	6	,	,	PUNCT
cana-1747	257	7	trigonometrical	trigonometrical	ADJ
cana-1747	257	8	series	series	NOUN
cana-1747	257	9	,	,	PUNCT
cana-1747	257	10	cambridge	cambridge	PROPN
cana-1747	257	11	university	university	PROPN
cana-1747	257	12	press	press	PROPN
cana-1747	257	13	,	,	PUNCT
cana-1747	257	14	cambridge	cambridge	PROPN
cana-1747	257	15	,	,	PUNCT
cana-1747	257	16	1959	1959	NUM
cana-1747	257	17	.	.	PUNCT
cana-1747	258	1	[	[	X
cana-1747	258	2	3	3	X
cana-1747	258	3	]	]	X
cana-1747	258	4	h.l	h.l	PROPN
cana-1747	258	5	.	.	PROPN
cana-1747	258	6	royden	royden	PROPN
cana-1747	258	7	and	and	CCONJ
cana-1747	258	8	p.m.	p.m.	NOUN
cana-1747	258	9	fitzpatrick	fitzpatrick	PROPN
cana-1747	258	10	,	,	PUNCT
cana-1747	258	11	real	real	ADJ
cana-1747	258	12	analysis	analysis	NOUN
cana-1747	258	13	,	,	PUNCT
cana-1747	258	14	fourth	fourth	ADJ
cana-1747	258	15	edition	edition	NOUN
cana-1747	258	16	,	,	PUNCT
cana-1747	258	17	prentice	prentice	NOUN
cana-1747	258	18	hall	hall	NOUN
cana-1747	258	19	of	of	ADP
cana-1747	258	20	india	india	PROPN
cana-1747	258	21	,	,	PUNCT
cana-1747	258	22	2010	2010	NUM
cana-1747	258	23	.	.	PUNCT
cana-1747	259	1	[	[	X
cana-1747	259	2	4	4	X
cana-1747	259	3	]	]	PUNCT
cana-1747	259	4	m.	m.	NOUN
cana-1747	259	5	weiss	weiss	PROPN
cana-1747	259	6	,	,	PUNCT
cana-1747	259	7	the	the	DET
cana-1747	259	8	law	law	NOUN
cana-1747	259	9	of	of	ADP
cana-1747	259	10	the	the	DET
cana-1747	259	11	iterated	iterated	ADJ
cana-1747	259	12	logarithm	logarithm	NOUN
cana-1747	259	13	for	for	ADP
cana-1747	259	14	lacunary	lacunary	ADJ
cana-1747	259	15	trigonometric	trigonometric	ADJ
cana-1747	259	16	series	series	NOUN
cana-1747	259	17	,	,	PUNCT
cana-1747	259	18	transaction	transaction	NOUN
cana-1747	259	19	of	of	ADP
cana-1747	259	20	american	american	PROPN
cana-1747	259	21	mathematical	mathematical	PROPN
cana-1747	259	22	society	society	NOUN
cana-1747	259	23	,	,	PUNCT
cana-1747	259	24	91	91	NUM
cana-1747	259	25	(	(	PUNCT
cana-1747	259	26	1959	1959	NUM
cana-1747	259	27	)	)	PUNCT
cana-1747	259	28	,	,	PUNCT
cana-1747	259	29	444	444	NUM
cana-1747	259	30	-	-	SYM
cana-1747	259	31	469	469	NUM
cana-1747	259	32	.	.	PUNCT
cana-1747	260	1	[	[	X
cana-1747	260	2	5	5	NUM
cana-1747	260	3	]	]	X
cana-1747	260	4	n.	n.	PROPN
cana-1747	260	5	kolmogorov	kolmogorov	PROPN
cana-1747	260	6	,	,	PUNCT
cana-1747	260	7	uber	uber	PROPN
cana-1747	260	8	des	des	PROPN
cana-1747	260	9	geset	geset	PROPN
cana-1747	260	10	:	:	PUNCT
cana-1747	260	11	des	des	X
cana-1747	260	12	iterierten	iterierten	PROPN
cana-1747	260	13	logarithmus	logarithmus	ADV
cana-1747	260	14	,	,	PUNCT
cana-1747	260	15	mathematische	mathematische	NOUN
cana-1747	260	16	annalen	annalen	VERB
cana-1747	260	17	101	101	NUM
cana-1747	260	18	(	(	PUNCT
cana-1747	260	19	1929	1929	NUM
cana-1747	260	20	)	)	PUNCT
cana-1747	260	21	,	,	PUNCT
cana-1747	260	22	136	136	NUM
cana-1747	260	23	-	-	SYM
cana-1747	260	24	139	139	NUM
cana-1747	260	25	.	.	PUNCT
cana-1747	261	1	[	[	X
cana-1747	261	2	6	6	NUM
cana-1747	261	3	]	]	PUNCT
cana-1747	261	4	p.	p.	NOUN
cana-1747	261	5	erdos	erdo	NOUN
cana-1747	261	6	and	and	CCONJ
cana-1747	261	7	i.s	i.s	PROPN
cana-1747	261	8	.	.	PROPN
cana-1747	261	9	gal	gal	PROPN
cana-1747	261	10	,	,	PUNCT
cana-1747	261	11	on	on	ADP
cana-1747	261	12	the	the	DET
cana-1747	261	13	law	law	NOUN
cana-1747	261	14	of	of	ADP
cana-1747	261	15	the	the	DET
cana-1747	261	16	iterated	iterated	ADJ
cana-1747	261	17	logarithm	logarithm	NOUN
cana-1747	261	18	,	,	PUNCT
cana-1747	261	19	proceedings	proceeding	NOUN
cana-1747	261	20	of	of	ADP
cana-1747	261	21	the	the	DET
cana-1747	261	22	koninklijke	koninklijke	PROPN
cana-1747	261	23	nederlandse	nederlandse	PROPN
cana-1747	261	24	akademie	akademie	PROPN
cana-1747	261	25	van	van	PROPN
cana-1747	261	26	wetenschappen	wetenschappen	PROPN
cana-1747	261	27	a	a	PRON
cana-1747	261	28	,	,	PUNCT
cana-1747	261	29	58	58	NUM
cana-1747	261	30	(	(	PUNCT
cana-1747	261	31	1955	1955	NUM
cana-1747	261	32	)	)	PUNCT
cana-1747	261	33	,	,	PUNCT
cana-1747	261	34	65	65	NUM
cana-1747	261	35	-	-	SYM
cana-1747	261	36	84	84	NUM
cana-1747	261	37	.	.	PUNCT
cana-1747	262	1	[	[	X
cana-1747	262	2	7	7	X
cana-1747	262	3	]	]	X
cana-1747	262	4	r.	r.	NOUN
cana-1747	262	5	banelos	banelos	PROPN
cana-1747	262	6	and	and	CCONJ
cana-1747	262	7	c.	c.	PROPN
cana-1747	262	8	n.	n.	PROPN
cana-1747	262	9	moore	moore	PROPN
cana-1747	262	10	,	,	PUNCT
cana-1747	262	11	probabilistic	probabilistic	ADJ
cana-1747	262	12	behaviour	behaviour	NOUN
cana-1747	262	13	of	of	ADP
cana-1747	262	14	harmonic	harmonic	ADJ
cana-1747	262	15	functions	function	NOUN
cana-1747	262	16	,	,	PUNCT
cana-1747	262	17	birkhasaur	birkhasaur	NOUN
cana-1747	262	18	verlag	verlag	PROPN
cana-1747	262	19	,	,	PUNCT
cana-1747	262	20	1991	1991	NUM
cana-1747	262	21	.	.	PUNCT
cana-1747	263	1	communications	communication	NOUN
cana-1747	263	2	on	on	ADP
cana-1747	263	3	applied	apply	VERB
cana-1747	263	4	nonlinear	nonlinear	ADJ
cana-1747	263	5	analysis	analysis	NOUN
cana-1747	263	6	issn	issn	NOUN
cana-1747	263	7	:	:	PUNCT
cana-1747	263	8	1074	1074	NUM
cana-1747	263	9	-	-	PUNCT
cana-1747	263	10	133x	133x	NUM
cana-1747	263	11	vol	vol	NOUN
cana-1747	263	12	32	32	NUM
cana-1747	263	13	no	no	NOUN
cana-1747	263	14	.	.	NOUN
cana-1747	263	15	2	2	NUM
cana-1747	263	16	(	(	PUNCT
cana-1747	263	17	2025	2025	NUM
cana-1747	263	18	)	)	PUNCT
cana-1747	263	19	366	366	NUM
cana-1747	263	20	https://internationalpubls.com	https://internationalpubls.com	X
cana-1747	264	1	[	[	X
cana-1747	264	2	8	8	NUM
cana-1747	264	3	]	]	X
cana-1747	264	4	r.	r.	PROPN
cana-1747	264	5	salem	salem	PROPN
cana-1747	264	6	and	and	CCONJ
cana-1747	264	7	a.	a.	NOUN
cana-1747	264	8	zygmund	zygmund	PROPN
cana-1747	264	9	,	,	PUNCT
cana-1747	264	10	la	la	PROPN
cana-1747	264	11	loi	loi	PROPN
cana-1747	264	12	du	du	PROPN
cana-1747	264	13	logarithme	logarithme	PROPN
cana-1747	264	14	itre	itre	PROPN
cana-1747	264	15	pour	pour	PROPN
cana-1747	264	16	les	les	PROPN
cana-1747	264	17	series	series	PROPN
cana-1747	264	18	trigonometriques	trigonometrique	NOUN
cana-1747	264	19	lacunaires	lacunaire	NOUN
cana-1747	264	20	,	,	PUNCT
cana-1747	264	21	bulletin	bulletin	PROPN
cana-1747	264	22	des	des	PROPN
cana-1747	264	23	sciences	sciences	PROPN
cana-1747	264	24	mathématiques	mathématique	NOUN
cana-1747	264	25	,	,	PUNCT
cana-1747	264	26	74	74	NUM
cana-1747	264	27	(	(	PUNCT
cana-1747	264	28	1950	1950	NUM
cana-1747	264	29	)	)	PUNCT
cana-1747	264	30	,	,	PUNCT
cana-1747	264	31	209	209	NUM
cana-1747	264	32	-	-	SYM
cana-1747	264	33	224	224	NUM
cana-1747	264	34	.	.	PUNCT
cana-1747	265	1	[	[	X
cana-1747	265	2	9	9	NUM
cana-1747	265	3	]	]	PUNCT
cana-1747	265	4	s.	s.	PROPN
cana-1747	265	5	ghimire	ghimire	PROPN
cana-1747	265	6	and	and	CCONJ
cana-1747	265	7	c.n	c.n	PROPN
cana-1747	265	8	.	.	PROPN
cana-1747	265	9	moore	moore	PROPN
cana-1747	265	10	,	,	PUNCT
cana-1747	265	11	a	a	DET
cana-1747	265	12	lower	lower	ADV
cana-1747	265	13	bound	bind	VERB
cana-1747	265	14	in	in	ADP
cana-1747	265	15	the	the	DET
cana-1747	265	16	tail	tail	NOUN
cana-1747	265	17	law	law	NOUN
cana-1747	265	18	of	of	ADP
cana-1747	265	19	the	the	DET
cana-1747	265	20	iterated	iterated	ADJ
cana-1747	265	21	logarithm	logarithm	NOUN
cana-1747	265	22	for	for	ADP
cana-1747	265	23	lacunary	lacunary	ADJ
cana-1747	265	24	trigonometric	trigonometric	ADJ
cana-1747	265	25	series	series	NOUN
cana-1747	265	26	,	,	PUNCT
cana-1747	265	27	proceeding	proceed	VERB
cana-1747	265	28	of	of	ADP
cana-1747	265	29	american	american	PROPN
cana-1747	265	30	mathematical	mathematical	PROPN
cana-1747	265	31	society	society	NOUN
cana-1747	265	32	,	,	PUNCT
cana-1747	265	33	142(2014	142(2014	NUM
cana-1747	265	34	)	)	PUNCT
cana-1747	265	35	,	,	PUNCT
cana-1747	265	36	3207	3207	NUM
cana-1747	265	37	-	-	SYM
cana-1747	265	38	3216	3216	NUM
cana-1747	265	39	.	.	PUNCT
cana-1747	266	1	[	[	X
cana-1747	266	2	10	10	NUM
cana-1747	266	3	]	]	X
cana-1747	266	4	s.	s.	PROPN
cana-1747	266	5	ghimire	ghimire	PROPN
cana-1747	266	6	,	,	PUNCT
cana-1747	266	7	one	one	NUM
cana-1747	266	8	-	-	PUNCT
cana-1747	266	9	sided	sided	ADJ
cana-1747	266	10	law	law	NOUN
cana-1747	266	11	of	of	ADP
cana-1747	266	12	the	the	DET
cana-1747	266	13	iterated	iterated	ADJ
cana-1747	266	14	logarithm	logarithm	NOUN
cana-1747	266	15	for	for	ADP
cana-1747	266	16	dyadic	dyadic	ADJ
cana-1747	266	17	martingale	martingale	NOUN
cana-1747	266	18	using	use	VERB
cana-1747	266	19	sub	sub	ADJ
cana-1747	266	20	-	-	ADJ
cana-1747	266	21	gaussian	gaussian	ADJ
cana-1747	266	22	estimates	estimate	NOUN
cana-1747	266	23	,	,	PUNCT
cana-1747	266	24	open	open	ADJ
cana-1747	266	25	journal	journal	NOUN
cana-1747	266	26	of	of	ADP
cana-1747	266	27	mathematical	mathematical	ADJ
cana-1747	266	28	analysis	analysis	NOUN
cana-1747	266	29	,	,	PUNCT
cana-1747	266	30	6(1	6(1	NUM
cana-1747	266	31	)	)	PUNCT
cana-1747	266	32	(	(	PUNCT
cana-1747	266	33	2022	2022	NUM
cana-1747	266	34	)	)	PUNCT
cana-1747	266	35	,	,	PUNCT
cana-1747	266	36	1	1	NUM
cana-1747	266	37	-	-	SYM
cana-1747	266	38	6	6	NUM
cana-1747	266	39	.	.	PUNCT
cana-1747	267	1	[	[	X
cana-1747	267	2	11	11	NUM
cana-1747	267	3	]	]	PUNCT
cana-1747	267	4	s.	s.	PROPN
cana-1747	267	5	ghimire	ghimire	PROPN
cana-1747	267	6	,	,	PUNCT
cana-1747	267	7	one	one	NUM
cana-1747	267	8	-	-	PUNCT
cana-1747	267	9	sided	sided	ADJ
cana-1747	267	10	version	version	NOUN
cana-1747	267	11	of	of	ADP
cana-1747	267	12	the	the	DET
cana-1747	267	13	law	law	NOUN
cana-1747	267	14	of	of	ADP
cana-1747	267	15	the	the	DET
cana-1747	267	16	iterated	iterated	ADJ
cana-1747	267	17	logarithm	logarithm	NOUN
cana-1747	267	18	for	for	ADP
cana-1747	267	19	summations	summation	NOUN
cana-1747	267	20	of	of	ADP
cana-1747	267	21	signum	signum	PROPN
cana-1747	267	22	functions	function	NOUN
cana-1747	267	23	,	,	PUNCT
cana-1747	267	24	journal	journal	NOUN
cana-1747	267	25	of	of	ADP
cana-1747	267	26	mathematics	mathematic	NOUN
cana-1747	267	27	,	,	PUNCT
cana-1747	267	28	2023	2023	NUM
cana-1747	267	29	(	(	PUNCT
cana-1747	267	30	2023	2023	NUM
cana-1747	267	31	)	)	PUNCT
cana-1747	267	32	,	,	PUNCT
cana-1747	267	33	1	1	NUM
cana-1747	267	34	-	-	SYM
cana-1747	267	35	7	7	NUM
cana-1747	267	36	.	.	PUNCT
cana-1747	268	1	[	[	X
cana-1747	268	2	12	12	NUM
cana-1747	268	3	]	]	PUNCT
cana-1747	268	4	w.	w.	PROPN
cana-1747	268	5	f.	f.	PROPN
cana-1747	268	6	stout	stout	PROPN
cana-1747	268	7	,	,	PUNCT
cana-1747	268	8	a	a	DET
cana-1747	268	9	martingale	martingale	ADJ
cana-1747	268	10	analogue	analogue	NOUN
cana-1747	268	11	of	of	ADP
cana-1747	268	12	kolmogorov	kolmogorov	PROPN
cana-1747	268	13	's	's	PART
cana-1747	268	14	law	law	NOUN
cana-1747	268	15	of	of	ADP
cana-1747	268	16	the	the	DET
cana-1747	268	17	iterated	iterated	ADJ
cana-1747	268	18	logarithm	logarithm	NOUN
cana-1747	268	19	,	,	PUNCT
cana-1747	268	20	zeitschriftfr	zeitschriftfr	PROPN
cana-1747	268	21	wahrscheinlichkeitstheoire	wahrscheinlichkeitstheoire	VERB
cana-1747	268	22	und	und	PROPN
cana-1747	268	23	verwandte	verwandte	PROPN
cana-1747	268	24	gebiete	gebiete	NOUN
cana-1747	268	25	,	,	PUNCT
cana-1747	268	26	15(4)(1970	15(4)(1970	NUM
cana-1747	268	27	)	)	PUNCT
cana-1747	268	28	,	,	PUNCT
cana-1747	268	29	279	279	NUM
cana-1747	268	30	-	-	SYM
cana-1747	268	31	290	290	NUM
cana-1747	268	32	.	.	PUNCT
cana-1747	269	1	https://www.sciencedirect.com/journal/bulletin-des-sciences-mathematiques	https://www.sciencedirect.com/journal/bulletin-des-sciences-mathematique	NOUN
cana-1747	269	2	https://www.sciencedirect.com/journal/bulletin-des-sciences-mathematiques	https://www.sciencedirect.com/journal/bulletin-des-sciences-mathematique	NOUN