id	sid	tid	token	lemma	pos
cana-5534	1	1	communications	communication	NOUN
cana-5534	1	2	on	on	ADP
cana-5534	1	3	applied	apply	VERB
cana-5534	1	4	nonlinear	nonlinear	ADJ
cana-5534	1	5	analysis	analysis	NOUN
cana-5534	1	6	issn	issn	NOUN
cana-5534	1	7	:	:	PUNCT
cana-5534	1	8	1074	1074	NUM
cana-5534	1	9	-	-	PUNCT
cana-5534	1	10	133x	133x	NUM
cana-5534	1	11	vol	vol	VERB
cana-5534	1	12	32	32	NUM
cana-5534	1	13	no	no	NOUN
cana-5534	1	14	.	.	PUNCT
cana-5534	2	1	10s	10	NOUN
cana-5534	2	2	(	(	PUNCT
cana-5534	2	3	2025	2025	NUM
cana-5534	2	4	)	)	PUNCT
cana-5534	2	5	2573	2573	NUM
cana-5534	2	6	https://internationalpubls.com	https://internationalpubls.com	NUM
cana-5534	2	7	generalized	generalized	ADJ
cana-5534	2	8	markov	markov	NOUN
cana-5534	2	9	inequality	inequality	NOUN
cana-5534	2	10	:	:	PUNCT
cana-5534	2	11	extensions	extension	NOUN
cana-5534	2	12	,	,	PUNCT
cana-5534	2	13	numerical	numerical	ADJ
cana-5534	2	14	illustrations	illustration	NOUN
cana-5534	2	15	,	,	PUNCT
cana-5534	2	16	and	and	CCONJ
cana-5534	2	17	multivariate	multivariate	VERB
cana-5534	2	18	chernoff	chernoff	PROPN
cana-5534	2	19	bounds	bound	NOUN
cana-5534	2	20	nouara	nouara	PROPN
cana-5534	2	21	lazri1,2	lazri1,2	PROPN
cana-5534	2	22	,	,	PUNCT
cana-5534	2	23	ahlem	ahlem	NOUN
cana-5534	2	24	djebar2	djebar2	PROPN
cana-5534	3	1	1higher	1higher	NUM
cana-5534	3	2	school	school	NOUN
cana-5534	3	3	of	of	ADP
cana-5534	3	4	management	management	PROPN
cana-5534	3	5	sciences	sciences	PROPN
cana-5534	3	6	annaba	annaba	PROPN
cana-5534	3	7	,	,	PUNCT
cana-5534	3	8	algeria	algeria	PROPN
cana-5534	3	9	lazri.nouara@essg-annaba.dz	lazri.nouara@essg-annaba.dz	PROPN
cana-5534	3	10	2laps	2laps	NUM
cana-5534	3	11	laboratory	laboratory	NOUN
cana-5534	3	12	,	,	PUNCT
cana-5534	3	13	badji	badji	PROPN
cana-5534	3	14	mokhtar	mokhtar	PROPN
cana-5534	3	15	-annaba	-annaba	PROPN
cana-5534	3	16	university	university	PROPN
cana-5534	3	17	,	,	PUNCT
cana-5534	3	18	box	box	NOUN
cana-5534	3	19	12	12	NUM
cana-5534	3	20	,	,	PUNCT
cana-5534	3	21	annaba	annaba	PROPN
cana-5534	3	22	,	,	PUNCT
cana-5534	3	23	23000	23000	NUM
cana-5534	3	24	algeria	algeria	PROPN
cana-5534	3	25	ahlem.djebar@univ-annaba.dz	ahlem.djebar@univ-annaba.dz	PROPN
cana-5534	3	26	article	article	NOUN
cana-5534	3	27	history	history	NOUN
cana-5534	3	28	:	:	PUNCT
cana-5534	3	29	received	receive	VERB
cana-5534	3	30	:	:	PUNCT
cana-5534	3	31	12	12	NUM
cana-5534	3	32	-	-	SYM
cana-5534	3	33	01	01	NUM
cana-5534	3	34	-	-	PUNCT
cana-5534	3	35	2025	2025	NUM
cana-5534	3	36	revised	revise	VERB
cana-5534	3	37	:	:	PUNCT
cana-5534	3	38	15	15	NUM
cana-5534	3	39	-	-	NUM
cana-5534	3	40	02	02	NUM
cana-5534	3	41	-	-	PUNCT
cana-5534	3	42	2025	2025	NUM
cana-5534	3	43	accepted	accept	VERB
cana-5534	3	44	:	:	PUNCT
cana-5534	3	45	01	01	NUM
cana-5534	3	46	-	-	SYM
cana-5534	3	47	03	03	NUM
cana-5534	3	48	-	-	PUNCT
cana-5534	3	49	2025	2025	NUM
cana-5534	3	50	abstract	abstract	NOUN
cana-5534	3	51	:	:	PUNCT
cana-5534	3	52	the	the	DET
cana-5534	3	53	classical	classical	ADJ
cana-5534	3	54	markov	markov	NOUN
cana-5534	3	55	inequality	inequality	NOUN
cana-5534	3	56	provides	provide	VERB
cana-5534	3	57	a	a	DET
cana-5534	3	58	simple	simple	ADJ
cana-5534	3	59	yet	yet	ADV
cana-5534	3	60	powerful	powerful	ADJ
cana-5534	3	61	bound	bind	VERB
cana-5534	3	62	on	on	ADP
cana-5534	3	63	tail	tail	NOUN
cana-5534	3	64	probabilities	probability	NOUN
cana-5534	3	65	of	of	ADP
cana-5534	3	66	non	non	ADJ
cana-5534	3	67	-	-	ADJ
cana-5534	3	68	negative	negative	ADJ
cana-5534	3	69	random	random	ADJ
cana-5534	3	70	variables	variable	NOUN
cana-5534	3	71	.	.	PUNCT
cana-5534	4	1	in	in	ADP
cana-5534	4	2	this	this	DET
cana-5534	4	3	paper	paper	NOUN
cana-5534	4	4	,	,	PUNCT
cana-5534	4	5	we	we	PRON
cana-5534	4	6	explore	explore	VERB
cana-5534	4	7	a	a	DET
cana-5534	4	8	generalization	generalization	NOUN
cana-5534	4	9	of	of	ADP
cana-5534	4	10	the	the	DET
cana-5534	4	11	markov	markov	NOUN
cana-5534	4	12	inequality	inequality	NOUN
cana-5534	4	13	that	that	PRON
cana-5534	4	14	employs	employ	VERB
cana-5534	4	15	convex	convex	PROPN
cana-5534	4	16	,	,	PUNCT
cana-5534	4	17	non	non	ADJ
cana-5534	4	18	-	-	ADJ
cana-5534	4	19	decreasing	decrease	VERB
cana-5534	4	20	functions	function	NOUN
cana-5534	4	21	to	to	PART
cana-5534	4	22	yield	yield	VERB
cana-5534	4	23	more	more	ADV
cana-5534	4	24	flexible	flexible	ADJ
cana-5534	4	25	and	and	CCONJ
cana-5534	4	26	often	often	ADV
cana-5534	4	27	tighter	tight	ADJ
cana-5534	4	28	bounds	bound	NOUN
cana-5534	4	29	.	.	PUNCT
cana-5534	5	1	we	we	PRON
cana-5534	5	2	present	present	VERB
cana-5534	5	3	three	three	NUM
cana-5534	5	4	distinct	distinct	ADJ
cana-5534	5	5	proofs	proof	NOUN
cana-5534	5	6	of	of	ADP
cana-5534	5	7	the	the	DET
cana-5534	5	8	generalized	generalized	ADJ
cana-5534	5	9	inequality	inequality	NOUN
cana-5534	5	10	,	,	PUNCT
cana-5534	5	11	including	include	VERB
cana-5534	5	12	one	one	NUM
cana-5534	5	13	based	base	VERB
cana-5534	5	14	on	on	ADP
cana-5534	5	15	jensen	jensen	PROPN
cana-5534	5	16	’s	’s	PART
cana-5534	5	17	inequality	inequality	NOUN
cana-5534	5	18	.	.	PUNCT
cana-5534	6	1	several	several	ADJ
cana-5534	6	2	convex	convex	NOUN
cana-5534	6	3	functions	function	NOUN
cana-5534	6	4	such	such	ADJ
cana-5534	6	5	as	as	ADP
cana-5534	6	6	𝜙(𝑥	𝜙(𝑥	NOUN
cana-5534	6	7	)	)	PUNCT
cana-5534	6	8	=	=	SYM
cana-5534	6	9	𝑥2	𝑥2	NOUN
cana-5534	6	10	and	and	CCONJ
cana-5534	6	11	ϕ(x	ϕ(x	NOUN
cana-5534	6	12	)	)	PUNCT
cana-5534	7	1	=	=	PUNCT
cana-5534	7	2	𝑒𝜆𝑥	𝑒𝜆𝑥	NOUN
cana-5534	7	3	are	be	AUX
cana-5534	7	4	examined	examine	VERB
cana-5534	7	5	,	,	PUNCT
cana-5534	7	6	and	and	CCONJ
cana-5534	7	7	their	their	PRON
cana-5534	7	8	impact	impact	NOUN
cana-5534	7	9	on	on	ADP
cana-5534	7	10	the	the	DET
cana-5534	7	11	tightness	tightness	NOUN
cana-5534	7	12	of	of	ADP
cana-5534	7	13	probabilistic	probabilistic	ADJ
cana-5534	7	14	bounds	bound	NOUN
cana-5534	7	15	is	be	AUX
cana-5534	7	16	illustrated	illustrate	VERB
cana-5534	7	17	through	through	ADP
cana-5534	7	18	detailed	detailed	ADJ
cana-5534	7	19	numerical	numerical	ADJ
cana-5534	7	20	examples	example	NOUN
cana-5534	7	21	.	.	PUNCT
cana-5534	8	1	furthermore	furthermore	ADV
cana-5534	8	2	,	,	PUNCT
cana-5534	8	3	we	we	PRON
cana-5534	8	4	extend	extend	VERB
cana-5534	8	5	the	the	DET
cana-5534	8	6	analysis	analysis	NOUN
cana-5534	8	7	to	to	ADP
cana-5534	8	8	the	the	DET
cana-5534	8	9	multivariate	multivariate	NOUN
cana-5534	8	10	setting	set	VERB
cana-5534	8	11	and	and	CCONJ
cana-5534	8	12	derive	derive	VERB
cana-5534	8	13	a	a	DET
cana-5534	8	14	version	version	NOUN
cana-5534	8	15	of	of	ADP
cana-5534	8	16	the	the	DET
cana-5534	8	17	chernoff	chernoff	NOUN
cana-5534	8	18	bound	bind	VERB
cana-5534	8	19	for	for	ADP
cana-5534	8	20	vector	vector	NOUN
cana-5534	8	21	-	-	PUNCT
cana-5534	8	22	valued	value	VERB
cana-5534	8	23	random	random	ADJ
cana-5534	8	24	variables	variable	NOUN
cana-5534	8	25	.	.	PUNCT
cana-5534	9	1	these	these	DET
cana-5534	9	2	results	result	NOUN
cana-5534	9	3	are	be	AUX
cana-5534	9	4	particularly	particularly	ADV
cana-5534	9	5	relevant	relevant	ADJ
cana-5534	9	6	in	in	ADP
cana-5534	9	7	areas	area	NOUN
cana-5534	9	8	such	such	ADJ
cana-5534	9	9	as	as	ADP
cana-5534	9	10	large	large	ADJ
cana-5534	9	11	deviations	deviation	NOUN
cana-5534	9	12	,	,	PUNCT
cana-5534	9	13	risk	risk	NOUN
cana-5534	9	14	theory	theory	NOUN
cana-5534	9	15	,	,	PUNCT
cana-5534	9	16	and	and	CCONJ
cana-5534	9	17	high	high	ADJ
cana-5534	9	18	-	-	PUNCT
cana-5534	9	19	dimensional	dimensional	ADJ
cana-5534	9	20	machine	machine	NOUN
cana-5534	9	21	learning	learning	NOUN
cana-5534	9	22	,	,	PUNCT
cana-5534	9	23	where	where	SCONJ
cana-5534	9	24	sharp	sharp	ADJ
cana-5534	9	25	tail	tail	NOUN
cana-5534	9	26	bounds	bound	NOUN
cana-5534	9	27	play	play	VERB
cana-5534	9	28	a	a	DET
cana-5534	9	29	critical	critical	ADJ
cana-5534	9	30	role	role	NOUN
cana-5534	9	31	.	.	PUNCT
cana-5534	10	1	theoretical	theoretical	ADJ
cana-5534	10	2	insights	insight	NOUN
cana-5534	10	3	are	be	AUX
cana-5534	10	4	complemented	complement	VERB
cana-5534	10	5	with	with	ADP
cana-5534	10	6	numerical	numerical	ADJ
cana-5534	10	7	illustrations	illustration	NOUN
cana-5534	10	8	to	to	PART
cana-5534	10	9	highlight	highlight	VERB
cana-5534	10	10	practical	practical	ADJ
cana-5534	10	11	implications	implication	NOUN
cana-5534	10	12	.	.	PUNCT
cana-5534	11	1	keywords	keyword	NOUN
cana-5534	11	2	:	:	PUNCT
cana-5534	11	3	markov	markov	NOUN
cana-5534	11	4	inequality	inequality	NOUN
cana-5534	11	5	;	;	PUNCT
cana-5534	11	6	chernoff	chernoff	NOUN
cana-5534	11	7	bound	bind	VERB
cana-5534	11	8	;	;	PUNCT
cana-5534	11	9	multivariate	multivariate	VERB
cana-5534	11	10	chernoff	chernoff	NOUN
cana-5534	11	11	bound	bind	VERB
cana-5534	11	12	;	;	PUNCT
cana-5534	11	13	portfolio	portfolio	NOUN
cana-5534	11	14	risk	risk	NOUN
cana-5534	11	15	assessment	assessment	NOUN
cana-5534	11	16	.	.	PUNCT
cana-5534	12	1	introduction	introduction	NOUN
cana-5534	12	2	probability	probability	NOUN
cana-5534	12	3	inequalities	inequality	NOUN
cana-5534	12	4	are	be	AUX
cana-5534	12	5	central	central	ADJ
cana-5534	12	6	to	to	ADP
cana-5534	12	7	the	the	DET
cana-5534	12	8	study	study	NOUN
cana-5534	12	9	of	of	ADP
cana-5534	12	10	stochastic	stochastic	ADJ
cana-5534	12	11	processes	process	NOUN
cana-5534	12	12	,	,	PUNCT
cana-5534	12	13	statistical	statistical	ADJ
cana-5534	12	14	inference	inference	NOUN
cana-5534	12	15	,	,	PUNCT
cana-5534	12	16	and	and	CCONJ
cana-5534	12	17	theoretical	theoretical	ADJ
cana-5534	12	18	computer	computer	NOUN
cana-5534	12	19	science	science	NOUN
cana-5534	12	20	.	.	PUNCT
cana-5534	13	1	among	among	ADP
cana-5534	13	2	the	the	DET
cana-5534	13	3	most	most	ADV
cana-5534	13	4	elementary	elementary	ADJ
cana-5534	13	5	and	and	CCONJ
cana-5534	13	6	powerful	powerful	ADJ
cana-5534	13	7	tools	tool	NOUN
cana-5534	13	8	is	be	AUX
cana-5534	13	9	the	the	DET
cana-5534	13	10	classical	classical	ADJ
cana-5534	13	11	markov	markov	NOUN
cana-5534	13	12	inequality	inequality	NOUN
cana-5534	13	13	,	,	PUNCT
cana-5534	13	14	which	which	PRON
cana-5534	13	15	provides	provide	VERB
cana-5534	13	16	a	a	DET
cana-5534	13	17	bound	bind	VERB
cana-5534	13	18	on	on	ADP
cana-5534	13	19	the	the	DET
cana-5534	13	20	probability	probability	NOUN
cana-5534	13	21	that	that	SCONJ
cana-5534	13	22	a	a	DET
cana-5534	13	23	non	non	ADJ
cana-5534	13	24	-	-	ADJ
cana-5534	13	25	negative	negative	ADJ
cana-5534	13	26	random	random	ADJ
cana-5534	13	27	variable	variable	NOUN
cana-5534	13	28	exceeds	exceed	VERB
cana-5534	13	29	a	a	DET
cana-5534	13	30	certain	certain	ADJ
cana-5534	13	31	threshold	threshold	NOUN
cana-5534	13	32	in	in	ADP
cana-5534	13	33	terms	term	NOUN
cana-5534	13	34	of	of	ADP
cana-5534	13	35	its	its	PRON
cana-5534	13	36	expected	expect	VERB
cana-5534	13	37	value	value	NOUN
cana-5534	13	38	.	.	PUNCT
cana-5534	14	1	specifically	specifically	ADV
cana-5534	14	2	,	,	PUNCT
cana-5534	14	3	if	if	SCONJ
cana-5534	14	4	x	x	PRON
cana-5534	14	5	is	be	AUX
cana-5534	14	6	a	a	DET
cana-5534	14	7	nonnegative	nonnegative	ADJ
cana-5534	14	8	random	random	ADJ
cana-5534	14	9	variable	variable	NOUN
cana-5534	14	10	and	and	CCONJ
cana-5534	14	11	a	a	DET
cana-5534	14	12	>	>	X
cana-5534	14	13	0	0	NUM
cana-5534	14	14	,	,	PUNCT
cana-5534	14	15	then	then	ADV
cana-5534	14	16	:	:	PUNCT
cana-5534	14	17	p(x	p(x	VERB
cana-5534	14	18	≥	≥	PRON
cana-5534	14	19	a	a	DET
cana-5534	14	20	)	)	PUNCT
cana-5534	14	21	≤	≤	NOUN
cana-5534	14	22	𝐸[𝑋	𝐸[𝑋	NOUN
cana-5534	14	23	]	]	PUNCT
cana-5534	15	1	𝑎	𝑎	X
cana-5534	15	2	this	this	DET
cana-5534	15	3	inequality	inequality	NOUN
cana-5534	15	4	is	be	AUX
cana-5534	15	5	a	a	DET
cana-5534	15	6	cornerstone	cornerstone	NOUN
cana-5534	15	7	of	of	ADP
cana-5534	15	8	probability	probability	NOUN
cana-5534	15	9	theory	theory	NOUN
cana-5534	15	10	and	and	CCONJ
cana-5534	15	11	forms	form	VERB
cana-5534	15	12	the	the	DET
cana-5534	15	13	basis	basis	NOUN
cana-5534	15	14	for	for	ADP
cana-5534	15	15	more	more	ADJ
cana-5534	15	16	refined	refined	ADJ
cana-5534	15	17	tools	tool	NOUN
cana-5534	15	18	such	such	ADJ
cana-5534	15	19	as	as	ADP
cana-5534	15	20	chebyshev	chebyshev	PROPN
cana-5534	15	21	’s	’s	PART
cana-5534	15	22	inequality	inequality	NOUN
cana-5534	15	23	,	,	PUNCT
cana-5534	15	24	chernoff	chernoff	NOUN
cana-5534	15	25	bounds	bound	NOUN
cana-5534	15	26	,	,	PUNCT
cana-5534	15	27	and	and	CCONJ
cana-5534	15	28	various	various	ADJ
cana-5534	15	29	concentration	concentration	NOUN
cana-5534	15	30	inequalities	inequality	NOUN
cana-5534	15	31	[	[	X
cana-5534	15	32	1	1	NUM
cana-5534	15	33	,	,	PUNCT
cana-5534	15	34	2	2	NUM
cana-5534	15	35	]	]	PUNCT
cana-5534	15	36	.	.	PUNCT
cana-5534	16	1	a	a	DET
cana-5534	16	2	natural	natural	ADJ
cana-5534	16	3	and	and	CCONJ
cana-5534	16	4	important	important	ADJ
cana-5534	16	5	generalization	generalization	NOUN
cana-5534	16	6	of	of	ADP
cana-5534	16	7	the	the	DET
cana-5534	16	8	markov	markov	NOUN
cana-5534	16	9	inequality	inequality	NOUN
cana-5534	16	10	involves	involve	VERB
cana-5534	16	11	replacing	replace	VERB
cana-5534	16	12	the	the	DET
cana-5534	16	13	identity	identity	NOUN
cana-5534	16	14	function	function	NOUN
cana-5534	16	15	x	x	PUNCT
cana-5534	16	16	↦	↦	PROPN
cana-5534	16	17	x	x	PUNCT
cana-5534	16	18	with	with	ADP
cana-5534	16	19	a	a	DET
cana-5534	16	20	non	non	ADJ
cana-5534	16	21	-	-	ADJ
cana-5534	16	22	decreasing	decrease	VERB
cana-5534	16	23	convex	convex	NOUN
cana-5534	16	24	function	function	NOUN
cana-5534	16	25	ϕ	ϕ	NOUN
cana-5534	16	26	,	,	PUNCT
cana-5534	16	27	leading	lead	VERB
cana-5534	16	28	to	to	ADP
cana-5534	16	29	the	the	DET
cana-5534	16	30	generalized	generalize	VERB
cana-5534	16	31	markov	markov	NOUN
cana-5534	16	32	inequality	inequality	NOUN
cana-5534	16	33	:	:	PUNCT
cana-5534	16	34	𝑃(𝑋	𝑃(𝑋	PROPN
cana-5534	16	35	≥	≥	PROPN
cana-5534	16	36	𝑎	𝑎	NOUN
cana-5534	16	37	)	)	PUNCT
cana-5534	16	38	≤	≤	NOUN
cana-5534	16	39	𝐸[𝜙(𝑋	𝐸[𝜙(𝑋	ADJ
cana-5534	16	40	)	)	PUNCT
cana-5534	16	41	]	]	PUNCT
cana-5534	16	42	𝜙(𝑎	𝜙(𝑎	PROPN
cana-5534	16	43	)	)	PUNCT
cana-5534	16	44	.	.	PUNCT
cana-5534	17	1	mailto:lazri.nouara@essg-annaba.dz	mailto:lazri.nouara@essg-annaba.dz	PROPN
cana-5534	17	2	mailto:ahlem.djebar@univ-annaba.dz	mailto:ahlem.djebar@univ-annaba.dz	PROPN
cana-5534	17	3	communications	communication	NOUN
cana-5534	17	4	on	on	ADP
cana-5534	17	5	applied	apply	VERB
cana-5534	17	6	nonlinear	nonlinear	ADJ
cana-5534	17	7	analysis	analysis	NOUN
cana-5534	17	8	issn	issn	NOUN
cana-5534	17	9	:	:	PUNCT
cana-5534	17	10	1074	1074	NUM
cana-5534	17	11	-	-	PUNCT
cana-5534	17	12	133x	133x	NUM
cana-5534	17	13	vol	vol	VERB
cana-5534	17	14	32	32	NUM
cana-5534	17	15	no	no	NOUN
cana-5534	17	16	.	.	PUNCT
cana-5534	18	1	10s	10	NOUN
cana-5534	18	2	(	(	PUNCT
cana-5534	18	3	2025	2025	NUM
cana-5534	18	4	)	)	PUNCT
cana-5534	18	5	2574	2574	NUM
cana-5534	18	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-5534	19	1	this	this	DET
cana-5534	19	2	form	form	NOUN
cana-5534	19	3	allows	allow	VERB
cana-5534	19	4	for	for	ADP
cana-5534	19	5	tighter	tight	ADJ
cana-5534	19	6	and	and	CCONJ
cana-5534	19	7	more	more	ADV
cana-5534	19	8	flexible	flexible	ADJ
cana-5534	19	9	bounds	bound	NOUN
cana-5534	19	10	and	and	CCONJ
cana-5534	19	11	has	have	AUX
cana-5534	19	12	been	be	AUX
cana-5534	19	13	extensively	extensively	ADV
cana-5534	19	14	used	use	VERB
cana-5534	19	15	in	in	ADP
cana-5534	19	16	areas	area	NOUN
cana-5534	19	17	such	such	ADJ
cana-5534	19	18	as	as	ADP
cana-5534	19	19	large	large	ADJ
cana-5534	19	20	deviations	deviation	NOUN
cana-5534	19	21	theory	theory	NOUN
cana-5534	19	22	[	[	X
cana-5534	19	23	3	3	NUM
cana-5534	19	24	]	]	PUNCT
cana-5534	19	25	,	,	PUNCT
cana-5534	19	26	risk	risk	NOUN
cana-5534	19	27	management	management	NOUN
cana-5534	20	1	[	[	X
cana-5534	20	2	5	5	NUM
cana-5534	20	3	]	]	PUNCT
cana-5534	20	4	,	,	PUNCT
cana-5534	20	5	and	and	CCONJ
cana-5534	20	6	information	information	NOUN
cana-5534	20	7	theory	theory	NOUN
cana-5534	20	8	[	[	X
cana-5534	20	9	4	4	NUM
cana-5534	20	10	]	]	PUNCT
cana-5534	20	11	.	.	PUNCT
cana-5534	21	1	the	the	DET
cana-5534	21	2	purpose	purpose	NOUN
cana-5534	21	3	of	of	ADP
cana-5534	21	4	this	this	DET
cana-5534	21	5	paper	paper	NOUN
cana-5534	21	6	is	be	AUX
cana-5534	21	7	multifold	multifold	ADJ
cana-5534	21	8	.	.	PUNCT
cana-5534	22	1	we	we	PRON
cana-5534	22	2	begin	begin	VERB
cana-5534	22	3	by	by	ADP
cana-5534	22	4	presenting	present	VERB
cana-5534	22	5	three	three	NUM
cana-5534	22	6	distinct	distinct	ADJ
cana-5534	22	7	proofs	proof	NOUN
cana-5534	22	8	of	of	ADP
cana-5534	22	9	the	the	DET
cana-5534	22	10	generalized	generalize	VERB
cana-5534	22	11	markov	markov	NOUN
cana-5534	22	12	inequality	inequality	NOUN
cana-5534	22	13	,	,	PUNCT
cana-5534	22	14	including	include	VERB
cana-5534	22	15	one	one	NUM
cana-5534	22	16	based	base	VERB
cana-5534	22	17	on	on	ADP
cana-5534	22	18	jensen	jensen	PROPN
cana-5534	22	19	’s	’s	PART
cana-5534	22	20	inequality	inequality	NOUN
cana-5534	22	21	and	and	CCONJ
cana-5534	22	22	others	other	NOUN
cana-5534	22	23	rooted	root	VERB
cana-5534	22	24	in	in	ADP
cana-5534	22	25	fundamental	fundamental	ADJ
cana-5534	22	26	properties	property	NOUN
cana-5534	22	27	of	of	ADP
cana-5534	22	28	convex	convex	NOUN
cana-5534	22	29	functions	function	NOUN
cana-5534	22	30	.	.	PUNCT
cana-5534	23	1	we	we	PRON
cana-5534	23	2	then	then	ADV
cana-5534	23	3	explore	explore	VERB
cana-5534	23	4	how	how	SCONJ
cana-5534	23	5	different	different	ADJ
cana-5534	23	6	choices	choice	NOUN
cana-5534	23	7	of	of	ADP
cana-5534	23	8	ϕ	ϕ	NOUN
cana-5534	23	9	,	,	PUNCT
cana-5534	23	10	such	such	ADJ
cana-5534	23	11	as	as	ADP
cana-5534	23	12	𝜙(𝑥	𝜙(𝑥	NOUN
cana-5534	23	13	)	)	PUNCT
cana-5534	23	14	=	=	SYM
cana-5534	23	15	𝑥2	𝑥2	NOUN
cana-5534	23	16	and	and	CCONJ
cana-5534	23	17	𝜙(𝑥	𝜙(𝑥	NUM
cana-5534	23	18	)	)	PUNCT
cana-5534	24	1	=	=	SYM
cana-5534	24	2	𝑒𝜆𝑥	𝑒𝜆𝑥	ADJ
cana-5534	24	3	,	,	PUNCT
cana-5534	24	4	yield	yield	NOUN
cana-5534	24	5	bounds	bound	NOUN
cana-5534	24	6	of	of	ADP
cana-5534	24	7	varying	vary	VERB
cana-5534	24	8	tightness	tightness	NOUN
cana-5534	24	9	.	.	PUNCT
cana-5534	25	1	these	these	PRON
cana-5534	25	2	are	be	AUX
cana-5534	25	3	illustrated	illustrate	VERB
cana-5534	25	4	with	with	ADP
cana-5534	25	5	detailed	detailed	ADJ
cana-5534	25	6	numerical	numerical	ADJ
cana-5534	25	7	examples	example	NOUN
cana-5534	25	8	to	to	PART
cana-5534	25	9	highlight	highlight	VERB
cana-5534	25	10	their	their	PRON
cana-5534	25	11	practical	practical	ADJ
cana-5534	25	12	implications	implication	NOUN
cana-5534	25	13	.	.	PUNCT
cana-5534	26	1	beyond	beyond	ADP
cana-5534	26	2	the	the	DET
cana-5534	26	3	univariate	univariate	ADJ
cana-5534	26	4	case	case	NOUN
cana-5534	26	5	,	,	PUNCT
cana-5534	26	6	we	we	PRON
cana-5534	26	7	extend	extend	VERB
cana-5534	26	8	the	the	DET
cana-5534	26	9	discussion	discussion	NOUN
cana-5534	26	10	to	to	ADP
cana-5534	26	11	the	the	DET
cana-5534	26	12	multivariate	multivariate	NOUN
cana-5534	26	13	setting	setting	NOUN
cana-5534	26	14	,	,	PUNCT
cana-5534	26	15	where	where	SCONJ
cana-5534	26	16	we	we	PRON
cana-5534	26	17	derive	derive	VERB
cana-5534	26	18	a	a	DET
cana-5534	26	19	multivariate	multivariate	NOUN
cana-5534	26	20	version	version	NOUN
cana-5534	26	21	of	of	ADP
cana-5534	26	22	the	the	DET
cana-5534	26	23	chernoff	chernoff	NOUN
cana-5534	26	24	bound	bind	VERB
cana-5534	26	25	using	use	VERB
cana-5534	26	26	convex	convex	NOUN
cana-5534	26	27	analysis	analysis	NOUN
cana-5534	26	28	and	and	CCONJ
cana-5534	26	29	optimization	optimization	NOUN
cana-5534	26	30	.	.	PUNCT
cana-5534	27	1	such	such	ADJ
cana-5534	27	2	bounds	bound	NOUN
cana-5534	27	3	are	be	AUX
cana-5534	27	4	essential	essential	ADJ
cana-5534	27	5	in	in	ADP
cana-5534	27	6	high	high	ADJ
cana-5534	27	7	dimensional	dimensional	ADJ
cana-5534	27	8	statistics	statistic	NOUN
cana-5534	27	9	and	and	CCONJ
cana-5534	27	10	machine	machine	NOUN
cana-5534	27	11	learning	learning	NOUN
cana-5534	27	12	,	,	PUNCT
cana-5534	27	13	particularly	particularly	ADV
cana-5534	27	14	in	in	ADP
cana-5534	27	15	the	the	DET
cana-5534	27	16	analysis	analysis	NOUN
cana-5534	27	17	of	of	ADP
cana-5534	27	18	generalization	generalization	NOUN
cana-5534	27	19	,	,	PUNCT
cana-5534	27	20	robustness	robustness	NOUN
cana-5534	27	21	,	,	PUNCT
cana-5534	27	22	and	and	CCONJ
cana-5534	27	23	large	large	ADJ
cana-5534	27	24	deviations	deviation	NOUN
cana-5534	27	25	of	of	ADP
cana-5534	27	26	vector	vector	NOUN
cana-5534	27	27	-	-	PUNCT
cana-5534	27	28	valued	value	VERB
cana-5534	27	29	random	random	ADJ
cana-5534	27	30	variables	variable	NOUN
cana-5534	27	31	.	.	PUNCT
cana-5534	28	1	statement	statement	NOUN
cana-5534	28	2	of	of	ADP
cana-5534	28	3	the	the	DET
cana-5534	28	4	generalized	generalize	VERB
cana-5534	28	5	markov	markov	NOUN
cana-5534	28	6	inequality	inequality	NOUN
cana-5534	28	7	proposition	proposition	NOUN
cana-5534	28	8	1	1	NUM
cana-5534	28	9	.	.	PUNCT
cana-5534	29	1	let	let	VERB
cana-5534	29	2	x	x	PRON
cana-5534	29	3	be	be	AUX
cana-5534	29	4	a	a	DET
cana-5534	29	5	non	non	ADJ
cana-5534	29	6	-	-	ADJ
cana-5534	29	7	negative	negative	ADJ
cana-5534	29	8	random	random	ADJ
cana-5534	29	9	variable	variable	NOUN
cana-5534	29	10	,	,	PUNCT
cana-5534	29	11	and	and	CCONJ
cana-5534	29	12	let	let	VERB
cana-5534	29	13	ϕ	ϕ	NOUN
cana-5534	29	14	:	:	PUNCT
cana-5534	29	15	[	[	X
cana-5534	29	16	0	0	NUM
cana-5534	29	17	,	,	PUNCT
cana-5534	29	18	∞	∞	PROPN
cana-5534	29	19	)	)	PUNCT
cana-5534	29	20	→	→	PUNCT
cana-5534	30	1	[	[	X
cana-5534	30	2	0	0	NUM
cana-5534	30	3	,	,	PUNCT
cana-5534	30	4	∞	∞	PROPN
cana-5534	30	5	)	)	PUNCT
cana-5534	30	6	be	be	AUX
cana-5534	30	7	a	a	DET
cana-5534	30	8	convex	convex	ADJ
cana-5534	30	9	and	and	CCONJ
cana-5534	30	10	non	non	ADJ
cana-5534	30	11	-	-	ADJ
cana-5534	30	12	decreasing	decrease	VERB
cana-5534	30	13	function	function	NOUN
cana-5534	30	14	.	.	PUNCT
cana-5534	31	1	then	then	ADV
cana-5534	31	2	,	,	PUNCT
cana-5534	31	3	for	for	ADP
cana-5534	31	4	any	any	DET
cana-5534	31	5	a	a	DET
cana-5534	31	6	>	>	X
cana-5534	31	7	0	0	NUM
cana-5534	31	8	such	such	ADJ
cana-5534	31	9	that	that	SCONJ
cana-5534	31	10	ϕ(a	ϕ(a	NOUN
cana-5534	31	11	)	)	PUNCT
cana-5534	31	12	>	>	X
cana-5534	32	1	0	0	NUM
cana-5534	32	2	,	,	PUNCT
cana-5534	32	3	the	the	DET
cana-5534	32	4	following	follow	VERB
cana-5534	32	5	inequality	inequality	NOUN
cana-5534	32	6	holds	hold	VERB
cana-5534	32	7	:	:	PUNCT
cana-5534	32	8	𝑃(𝑋	𝑃(𝑋	PROPN
cana-5534	32	9	≥	≥	PROPN
cana-5534	32	10	𝑎	𝑎	NOUN
cana-5534	32	11	)	)	PUNCT
cana-5534	32	12	≤	≤	NOUN
cana-5534	32	13	𝐸[𝜙(𝑋	𝐸[𝜙(𝑋	ADJ
cana-5534	32	14	)	)	PUNCT
cana-5534	32	15	]	]	PUNCT
cana-5534	32	16	𝜙(𝑎	𝜙(𝑎	PROPN
cana-5534	32	17	)	)	PUNCT
cana-5534	32	18	.	.	PUNCT
cana-5534	33	1	(	(	PUNCT
cana-5534	33	2	1	1	X
cana-5534	33	3	)	)	PUNCT
cana-5534	33	4	proof	proof	NOUN
cana-5534	33	5	1	1	NUM
cana-5534	33	6	:	:	PUNCT
cana-5534	33	7	using	use	VERB
cana-5534	33	8	jensen	jensen	PROPN
cana-5534	33	9	’s	’s	PART
cana-5534	33	10	inequality	inequality	NOUN
cana-5534	33	11	proposition	proposition	NOUN
cana-5534	33	12	2	2	NUM
cana-5534	33	13	.	.	PUNCT
cana-5534	34	1	under	under	ADP
cana-5534	34	2	the	the	DET
cana-5534	34	3	assumptions	assumption	NOUN
cana-5534	34	4	of	of	ADP
cana-5534	34	5	proposition	proposition	NOUN
cana-5534	34	6	1	1	NUM
cana-5534	34	7	,	,	PUNCT
cana-5534	34	8	the	the	DET
cana-5534	34	9	inequality	inequality	NOUN
cana-5534	34	10	can	can	AUX
cana-5534	34	11	be	be	AUX
cana-5534	34	12	proven	prove	VERB
cana-5534	34	13	via	via	ADP
cana-5534	34	14	jensen	jensen	PROPN
cana-5534	34	15	’s	’s	PART
cana-5534	34	16	inequality	inequality	NOUN
cana-5534	34	17	.	.	PUNCT
cana-5534	35	1	proof	proof	NOUN
cana-5534	35	2	.	.	PUNCT
cana-5534	36	1	let	let	VERB
cana-5534	36	2	i{𝑋≥𝑎	i{𝑋≥𝑎	NOUN
cana-5534	36	3	}	}	PUNCT
cana-5534	36	4	denote	denote	VERB
cana-5534	36	5	the	the	DET
cana-5534	36	6	indicator	indicator	NOUN
cana-5534	36	7	function	function	NOUN
cana-5534	36	8	of	of	ADP
cana-5534	36	9	the	the	DET
cana-5534	36	10	event	event	NOUN
cana-5534	36	11	{	{	PUNCT
cana-5534	36	12	x	x	X
cana-5534	36	13	≥	≥	X
cana-5534	36	14	a	a	PRON
cana-5534	36	15	}	}	PUNCT
cana-5534	36	16	.	.	PUNCT
cana-5534	37	1	then	then	ADV
cana-5534	37	2	:	:	PUNCT
cana-5534	37	3	𝐸[𝜙(𝑋	𝐸[𝜙(𝑋	ADJ
cana-5534	37	4	)	)	PUNCT
cana-5534	37	5	]	]	PUNCT
cana-5534	37	6	≥	≥	X
cana-5534	37	7	𝐸[𝜙(𝑋	𝐸[𝜙(𝑋	ADJ
cana-5534	37	8	)	)	PUNCT
cana-5534	37	9	·	·	PUNCT
cana-5534	37	10	i{𝑋≥𝑎	i{𝑋≥𝑎	NOUN
cana-5534	37	11	}	}	PUNCT
cana-5534	37	12	]	]	PUNCT
cana-5534	37	13	.	.	PUNCT
cana-5534	38	1	since	since	SCONJ
cana-5534	38	2	ϕ	ϕ	PROPN
cana-5534	38	3	is	be	AUX
cana-5534	38	4	non	non	ADJ
cana-5534	38	5	-	-	ADJ
cana-5534	38	6	decreasing	decrease	VERB
cana-5534	38	7	and	and	CCONJ
cana-5534	38	8	convex	convex	NOUN
cana-5534	38	9	,	,	PUNCT
cana-5534	38	10	on	on	ADP
cana-5534	38	11	the	the	DET
cana-5534	38	12	event	event	NOUN
cana-5534	38	13	{	{	PUNCT
cana-5534	38	14	x	x	X
cana-5534	38	15	≥	≥	X
cana-5534	38	16	a	a	X
cana-5534	38	17	}	}	PUNCT
cana-5534	38	18	,	,	PUNCT
cana-5534	38	19	we	we	PRON
cana-5534	38	20	have	have	AUX
cana-5534	38	21	ϕ(x	ϕ(x	X
cana-5534	38	22	)	)	PUNCT
cana-5534	38	23	≥	≥	NOUN
cana-5534	38	24	ϕ(a	ϕ(a	NOUN
cana-5534	38	25	)	)	PUNCT
cana-5534	38	26	,	,	PUNCT
cana-5534	38	27	hence	hence	ADV
cana-5534	38	28	:	:	PUNCT
cana-5534	38	29	𝐸[𝜙(𝑋	𝐸[𝜙(𝑋	PROPN
cana-5534	38	30	)	)	PUNCT
cana-5534	38	31	]	]	PUNCT
cana-5534	38	32	≥	≥	X
cana-5534	38	33	𝜙(𝑎	𝜙(𝑎	PROPN
cana-5534	38	34	)	)	PUNCT
cana-5534	38	35	·	·	PUNCT
cana-5534	38	36	𝑃(𝑋	𝑃(𝑋	PROPN
cana-5534	38	37	≥	≥	NUM
cana-5534	38	38	𝑎	𝑎	NOUN
cana-5534	38	39	)	)	PUNCT
cana-5534	38	40	.	.	PUNCT
cana-5534	39	1	dividing	divide	VERB
cana-5534	39	2	both	both	DET
cana-5534	39	3	sides	side	NOUN
cana-5534	39	4	by	by	ADP
cana-5534	39	5	ϕ(a	ϕ(a	NOUN
cana-5534	39	6	)	)	PUNCT
cana-5534	39	7	>	>	X
cana-5534	39	8	0	0	PUNCT
cana-5534	39	9	gives	give	VERB
cana-5534	39	10	the	the	DET
cana-5534	39	11	desired	desire	VERB
cana-5534	39	12	result	result	NOUN
cana-5534	39	13	:	:	PUNCT
cana-5534	39	14	𝑃(𝑋	𝑃(𝑋	PROPN
cana-5534	39	15	≥	≥	NUM
cana-5534	39	16	𝑎	𝑎	X
cana-5534	39	17	)	)	PUNCT
cana-5534	39	18	≤	≤	NOUN
cana-5534	39	19	𝐸[𝜙(𝑋	𝐸[𝜙(𝑋	ADJ
cana-5534	39	20	)	)	PUNCT
cana-5534	39	21	]	]	PUNCT
cana-5534	39	22	𝜙(𝑎	𝜙(𝑎	PROPN
cana-5534	39	23	)	)	PUNCT
cana-5534	39	24	.	.	PUNCT
cana-5534	40	1	□	□	PUNCT
cana-5534	40	2	proof	proof	NOUN
cana-5534	40	3	2	2	NUM
cana-5534	40	4	:	:	PUNCT
cana-5534	40	5	decomposition	decomposition	NOUN
cana-5534	40	6	via	via	ADP
cana-5534	40	7	conditioning	conditioning	NOUN
cana-5534	40	8	proposition	proposition	NOUN
cana-5534	40	9	3	3	NUM
cana-5534	40	10	.	.	PUNCT
cana-5534	41	1	the	the	DET
cana-5534	41	2	inequality	inequality	NOUN
cana-5534	41	3	in	in	ADP
cana-5534	41	4	proposition	proposition	NOUN
cana-5534	41	5	1	1	NUM
cana-5534	41	6	also	also	ADV
cana-5534	41	7	follows	follow	VERB
cana-5534	41	8	from	from	ADP
cana-5534	41	9	a	a	DET
cana-5534	41	10	conditional	conditional	ADJ
cana-5534	41	11	decomposition	decomposition	NOUN
cana-5534	41	12	of	of	ADP
cana-5534	41	13	expectation	expectation	NOUN
cana-5534	41	14	.	.	PUNCT
cana-5534	42	1	proof	proof	NOUN
cana-5534	42	2	.	.	PUNCT
cana-5534	43	1	decompose	decompose	VERB
cana-5534	43	2	the	the	DET
cana-5534	43	3	expectation	expectation	NOUN
cana-5534	43	4	over	over	ADP
cana-5534	43	5	disjoint	disjoint	ADJ
cana-5534	43	6	events	event	NOUN
cana-5534	43	7	:	:	PUNCT
cana-5534	43	8	communications	communication	NOUN
cana-5534	43	9	on	on	ADP
cana-5534	43	10	applied	apply	VERB
cana-5534	43	11	nonlinear	nonlinear	ADJ
cana-5534	43	12	analysis	analysis	NOUN
cana-5534	43	13	issn	issn	NOUN
cana-5534	43	14	:	:	PUNCT
cana-5534	43	15	1074	1074	NUM
cana-5534	43	16	-	-	PUNCT
cana-5534	43	17	133x	133x	NUM
cana-5534	43	18	vol	vol	VERB
cana-5534	43	19	32	32	NUM
cana-5534	43	20	no	no	NOUN
cana-5534	43	21	.	.	PUNCT
cana-5534	44	1	10s	10	NOUN
cana-5534	44	2	(	(	PUNCT
cana-5534	44	3	2025	2025	NUM
cana-5534	44	4	)	)	PUNCT
cana-5534	44	5	2575	2575	NUM
cana-5534	44	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-5534	44	7	𝐸[𝜙(𝑋	𝐸[𝜙(𝑋	ADJ
cana-5534	44	8	)	)	PUNCT
cana-5534	44	9	]	]	PUNCT
cana-5534	44	10	=	=	SYM
cana-5534	44	11	𝐸[𝜙(𝑋	𝐸[𝜙(𝑋	ADJ
cana-5534	44	12	)	)	PUNCT
cana-5534	44	13	|	|	ADV
cana-5534	44	14	𝑋	𝑋	NOUN
cana-5534	44	15	<	<	X
cana-5534	44	16	𝑎	𝑎	X
cana-5534	44	17	]	]	X
cana-5534	44	18	·	·	PUNCT
cana-5534	44	19	𝑃(𝑋	𝑃(𝑋	X
cana-5534	44	20	<	<	X
cana-5534	44	21	𝑎	𝑎	X
cana-5534	44	22	)	)	PUNCT
cana-5534	45	1	+	+	CCONJ
cana-5534	45	2	𝐸[𝜙(𝑋	𝐸[𝜙(𝑋	ADJ
cana-5534	45	3	)	)	PUNCT
cana-5534	45	4	|	|	ADV
cana-5534	45	5	𝑋	𝑋	NOUN
cana-5534	45	6	≥	≥	NUM
cana-5534	45	7	𝑎	𝑎	NOUN
cana-5534	45	8	]	]	X
cana-5534	45	9	·	·	PUNCT
cana-5534	45	10	𝑃(𝑋	𝑃(𝑋	PROPN
cana-5534	45	11	≥	≥	NUM
cana-5534	45	12	𝑎	𝑎	NOUN
cana-5534	45	13	)	)	PUNCT
cana-5534	45	14	.	.	PUNCT
cana-5534	46	1	since	since	SCONJ
cana-5534	46	2	ϕ	ϕ	PROPN
cana-5534	46	3	is	be	AUX
cana-5534	46	4	non	non	ADJ
cana-5534	46	5	-	-	ADJ
cana-5534	46	6	decreasing	decrease	VERB
cana-5534	46	7	and	and	CCONJ
cana-5534	46	8	x	x	X
cana-5534	46	9	≥	≥	NOUN
cana-5534	46	10	a	a	PRON
cana-5534	46	11	on	on	ADP
cana-5534	46	12	the	the	DET
cana-5534	46	13	second	second	ADJ
cana-5534	46	14	event	event	NOUN
cana-5534	46	15	,	,	PUNCT
cana-5534	46	16	𝐸[𝜙(𝑋	𝐸[𝜙(𝑋	ADJ
cana-5534	46	17	)	)	PUNCT
cana-5534	46	18	|	|	ADV
cana-5534	46	19	𝑋	𝑋	NOUN
cana-5534	46	20	≥	≥	NUM
cana-5534	46	21	𝑎	𝑎	NOUN
cana-5534	46	22	]	]	X
cana-5534	46	23	≥	≥	NOUN
cana-5534	46	24	𝜙(𝑎	𝜙(𝑎	PROPN
cana-5534	46	25	)	)	PUNCT
cana-5534	46	26	.	.	PUNCT
cana-5534	47	1	therefore	therefore	ADV
cana-5534	47	2	,	,	PUNCT
cana-5534	47	3	𝐸[𝜙(𝑋	𝐸[𝜙(𝑋	ADJ
cana-5534	47	4	)	)	PUNCT
cana-5534	47	5	]	]	PUNCT
cana-5534	47	6	≥	≥	X
cana-5534	47	7	𝜙(𝑎	𝜙(𝑎	PROPN
cana-5534	47	8	)	)	PUNCT
cana-5534	47	9	·	·	PUNCT
cana-5534	48	1	𝑃(𝑋	𝑃(𝑋	PROPN
cana-5534	48	2	≥	≥	NUM
cana-5534	48	3	𝑎	𝑎	NOUN
cana-5534	48	4	)	)	PUNCT
cana-5534	48	5	,	,	PUNCT
cana-5534	48	6	which	which	PRON
cana-5534	48	7	again	again	ADV
cana-5534	48	8	yields	yield	VERB
cana-5534	48	9	:	:	PUNCT
cana-5534	48	10	𝑃(𝑋	𝑃(𝑋	PROPN
cana-5534	48	11	≥	≥	PROPN
cana-5534	48	12	𝑎	𝑎	NOUN
cana-5534	48	13	)	)	PUNCT
cana-5534	48	14	≤	≤	NOUN
cana-5534	48	15	𝐸[𝜙(𝑋	𝐸[𝜙(𝑋	ADJ
cana-5534	48	16	)	)	PUNCT
cana-5534	48	17	]	]	PUNCT
cana-5534	48	18	𝜙(𝑎	𝜙(𝑎	PROPN
cana-5534	48	19	)	)	PUNCT
cana-5534	48	20	.	.	PUNCT
cana-5534	49	1	proof	proof	NOUN
cana-5534	49	2	3	3	NUM
cana-5534	49	3	:	:	PUNCT
cana-5534	49	4	direct	direct	ADJ
cana-5534	49	5	inequality	inequality	NOUN
cana-5534	49	6	via	via	ADP
cana-5534	49	7	indicator	indicator	NOUN
cana-5534	49	8	function	function	NOUN
cana-5534	49	9	proposition	proposition	NOUN
cana-5534	49	10	4	4	NUM
cana-5534	49	11	.	.	PUNCT
cana-5534	50	1	the	the	DET
cana-5534	50	2	inequality	inequality	NOUN
cana-5534	50	3	in	in	ADP
cana-5534	50	4	proposition	proposition	NOUN
cana-5534	50	5	1	1	NUM
cana-5534	50	6	can	can	AUX
cana-5534	50	7	also	also	ADV
cana-5534	50	8	be	be	AUX
cana-5534	50	9	obtained	obtain	VERB
cana-5534	50	10	using	use	VERB
cana-5534	50	11	a	a	DET
cana-5534	50	12	simple	simple	ADJ
cana-5534	50	13	bound	bind	VERB
cana-5534	50	14	on	on	ADP
cana-5534	50	15	ϕ(x	ϕ(x	NOUN
cana-5534	50	16	)	)	PUNCT
cana-5534	50	17	.	.	PUNCT
cana-5534	51	1	proof	proof	NOUN
cana-5534	51	2	.	.	PUNCT
cana-5534	52	1	we	we	PRON
cana-5534	52	2	directly	directly	ADV
cana-5534	52	3	observe	observe	VERB
cana-5534	52	4	that	that	SCONJ
cana-5534	52	5	:	:	PUNCT
cana-5534	52	6	𝜙(𝑋	𝜙(𝑋	NUM
cana-5534	52	7	)	)	PUNCT
cana-5534	52	8	≥	≥	PRON
cana-5534	52	9	𝜙(𝑎	𝜙(𝑎	PROPN
cana-5534	52	10	)	)	PUNCT
cana-5534	52	11	·	·	PUNCT
cana-5534	52	12	i{𝑋≥𝑎	i{𝑋≥𝑎	NOUN
cana-5534	52	13	}	}	PUNCT
cana-5534	52	14	,	,	PUNCT
cana-5534	52	15	since	since	SCONJ
cana-5534	52	16	ϕ	ϕ	PROPN
cana-5534	52	17	is	be	AUX
cana-5534	52	18	non	non	ADJ
cana-5534	52	19	-	-	ADJ
cana-5534	52	20	decreasing	decrease	VERB
cana-5534	52	21	.	.	PUNCT
cana-5534	53	1	taking	take	VERB
cana-5534	53	2	expectations	expectation	NOUN
cana-5534	53	3	:	:	PUNCT
cana-5534	53	4	𝐸[𝜙(𝑋	𝐸[𝜙(𝑋	ADJ
cana-5534	53	5	)	)	PUNCT
cana-5534	53	6	]	]	PUNCT
cana-5534	53	7	≥	≥	X
cana-5534	53	8	𝐸[𝜙(𝑎	𝐸[𝜙(𝑎	VERB
cana-5534	53	9	)	)	PUNCT
cana-5534	53	10	·	·	PUNCT
cana-5534	53	11	i{𝑋≥𝑎	i{𝑋≥𝑎	NOUN
cana-5534	53	12	}	}	PUNCT
cana-5534	53	13	]	]	PUNCT
cana-5534	53	14	=	=	SYM
cana-5534	53	15	𝜙(𝑎	𝜙(𝑎	PROPN
cana-5534	53	16	)	)	PUNCT
cana-5534	53	17	·	·	PUNCT
cana-5534	53	18	𝑃(𝑋	𝑃(𝑋	PROPN
cana-5534	53	19	≥	≥	NUM
cana-5534	53	20	𝑎	𝑎	NOUN
cana-5534	53	21	)	)	PUNCT
cana-5534	53	22	.	.	PUNCT
cana-5534	54	1	dividing	divide	VERB
cana-5534	54	2	both	both	DET
cana-5534	54	3	sides	side	NOUN
cana-5534	54	4	by	by	ADP
cana-5534	54	5	ϕ(a	ϕ(a	NOUN
cana-5534	54	6	)	)	PUNCT
cana-5534	54	7	gives	give	VERB
cana-5534	54	8	the	the	DET
cana-5534	54	9	required	require	VERB
cana-5534	54	10	inequality	inequality	NOUN
cana-5534	54	11	:	:	PUNCT
cana-5534	54	12	𝑃(𝑋	𝑃(𝑋	PROPN
cana-5534	54	13	≥	≥	PROPN
cana-5534	54	14	𝑎	𝑎	NOUN
cana-5534	54	15	)	)	PUNCT
cana-5534	54	16	≤	≤	NOUN
cana-5534	54	17	𝐸[𝜙(𝑋	𝐸[𝜙(𝑋	ADJ
cana-5534	54	18	)	)	PUNCT
cana-5534	54	19	]	]	PUNCT
cana-5534	54	20	𝜙(𝑎	𝜙(𝑎	PROPN
cana-5534	54	21	)	)	PUNCT
cana-5534	54	22	.	.	PUNCT
cana-5534	55	1	□	□	PUNCT
cana-5534	55	2	remark	remark	NOUN
cana-5534	55	3	.	.	PUNCT
cana-5534	56	1	each	each	PRON
cana-5534	56	2	of	of	ADP
cana-5534	56	3	the	the	DET
cana-5534	56	4	above	above	ADJ
cana-5534	56	5	proofs	proof	NOUN
cana-5534	56	6	highlights	highlight	NOUN
cana-5534	56	7	a	a	DET
cana-5534	56	8	different	different	ADJ
cana-5534	56	9	perspective	perspective	NOUN
cana-5534	56	10	:	:	PUNCT
cana-5534	56	11	jensen	jensen	PROPN
cana-5534	56	12	’s	’s	PART
cana-5534	56	13	inequality	inequality	NOUN
cana-5534	56	14	emphasizes	emphasize	VERB
cana-5534	56	15	convexity	convexity	NOUN
cana-5534	56	16	,	,	PUNCT
cana-5534	56	17	conditioning	conditioning	NOUN
cana-5534	56	18	showcases	showcase	VERB
cana-5534	56	19	probabilistic	probabilistic	ADJ
cana-5534	56	20	decomposition	decomposition	NOUN
cana-5534	56	21	,	,	PUNCT
cana-5534	56	22	and	and	CCONJ
cana-5534	56	23	the	the	DET
cana-5534	56	24	indicator	indicator	NOUN
cana-5534	56	25	function	function	NOUN
cana-5534	56	26	approach	approach	NOUN
cana-5534	56	27	provides	provide	VERB
cana-5534	56	28	a	a	DET
cana-5534	56	29	direct	direct	ADJ
cana-5534	56	30	algebraic	algebraic	ADJ
cana-5534	56	31	bound	bind	VERB
cana-5534	56	32	.	.	PUNCT
cana-5534	57	1	multivariate	multivariate	NOUN
cana-5534	57	2	chernoff	chernoff	PROPN
cana-5534	57	3	bound	bind	VERB
cana-5534	57	4	in	in	ADP
cana-5534	57	5	probability	probability	NOUN
cana-5534	57	6	theory	theory	NOUN
cana-5534	57	7	and	and	CCONJ
cana-5534	57	8	statistical	statistical	ADJ
cana-5534	57	9	learning	learning	NOUN
cana-5534	57	10	,	,	PUNCT
cana-5534	57	11	the	the	DET
cana-5534	57	12	chernoff	chernoff	NOUN
cana-5534	57	13	bound	bind	VERB
cana-5534	57	14	is	be	AUX
cana-5534	57	15	a	a	DET
cana-5534	57	16	powerful	powerful	ADJ
cana-5534	57	17	exponential	exponential	ADJ
cana-5534	57	18	inequality	inequality	NOUN
cana-5534	57	19	that	that	PRON
cana-5534	57	20	provides	provide	VERB
cana-5534	57	21	tight	tight	ADJ
cana-5534	57	22	upper	upper	ADJ
cana-5534	57	23	bounds	bound	NOUN
cana-5534	57	24	on	on	ADP
cana-5534	57	25	the	the	DET
cana-5534	57	26	tail	tail	NOUN
cana-5534	57	27	probabilities	probability	NOUN
cana-5534	57	28	of	of	ADP
cana-5534	57	29	sums	sum	NOUN
cana-5534	57	30	or	or	CCONJ
cana-5534	57	31	functions	function	NOUN
cana-5534	57	32	of	of	ADP
cana-5534	57	33	random	random	ADJ
cana-5534	57	34	variables	variable	NOUN
cana-5534	57	35	.	.	PUNCT
cana-5534	58	1	the	the	DET
cana-5534	58	2	classical	classical	ADJ
cana-5534	58	3	(	(	PUNCT
cana-5534	58	4	univariate	univariate	ADJ
cana-5534	58	5	)	)	PUNCT
cana-5534	58	6	chernoff	chernoff	NOUN
cana-5534	58	7	bound	bind	VERB
cana-5534	58	8	is	be	AUX
cana-5534	58	9	widely	widely	ADV
cana-5534	58	10	used	use	VERB
cana-5534	58	11	in	in	ADP
cana-5534	58	12	analyzing	analyze	VERB
cana-5534	58	13	the	the	DET
cana-5534	58	14	performance	performance	NOUN
cana-5534	58	15	of	of	ADP
cana-5534	58	16	randomized	randomized	ADJ
cana-5534	58	17	algorithms	algorithm	NOUN
cana-5534	58	18	,	,	PUNCT
cana-5534	58	19	information	information	NOUN
cana-5534	58	20	theory	theory	NOUN
cana-5534	58	21	,	,	PUNCT
cana-5534	58	22	and	and	CCONJ
cana-5534	58	23	risk	risk	NOUN
cana-5534	58	24	management	management	NOUN
cana-5534	58	25	.	.	PUNCT
cana-5534	59	1	in	in	ADP
cana-5534	59	2	multivariate	multivariate	NOUN
cana-5534	59	3	settings	setting	NOUN
cana-5534	59	4	,	,	PUNCT
cana-5534	59	5	the	the	DET
cana-5534	59	6	inequality	inequality	NOUN
cana-5534	59	7	generalizes	generalize	VERB
cana-5534	59	8	to	to	ADP
cana-5534	59	9	vectors	vector	NOUN
cana-5534	59	10	of	of	ADP
cana-5534	59	11	random	random	ADJ
cana-5534	59	12	variables	variable	NOUN
cana-5534	59	13	,	,	PUNCT
cana-5534	59	14	allowing	allow	VERB
cana-5534	59	15	for	for	ADP
cana-5534	59	16	simultaneous	simultaneous	ADJ
cana-5534	59	17	control	control	NOUN
cana-5534	59	18	of	of	ADP
cana-5534	59	19	tail	tail	NOUN
cana-5534	59	20	probabilities	probability	NOUN
cana-5534	59	21	in	in	ADP
cana-5534	59	22	multiple	multiple	ADJ
cana-5534	59	23	dimensions	dimension	NOUN
cana-5534	59	24	(	(	PUNCT
cana-5534	59	25	see	see	VERB
cana-5534	59	26	theorem	theorem	NOUN
cana-5534	59	27	1	1	NUM
cana-5534	59	28	)	)	PUNCT
cana-5534	59	29	.	.	PUNCT
cana-5534	60	1	communications	communication	NOUN
cana-5534	60	2	on	on	ADP
cana-5534	60	3	applied	apply	VERB
cana-5534	60	4	nonlinear	nonlinear	ADJ
cana-5534	60	5	analysis	analysis	NOUN
cana-5534	60	6	issn	issn	NOUN
cana-5534	60	7	:	:	PUNCT
cana-5534	60	8	1074	1074	NUM
cana-5534	60	9	-	-	PUNCT
cana-5534	60	10	133x	133x	NUM
cana-5534	60	11	vol	vol	VERB
cana-5534	60	12	32	32	NUM
cana-5534	60	13	no	no	NOUN
cana-5534	60	14	.	.	PUNCT
cana-5534	61	1	10s	10	NOUN
cana-5534	61	2	(	(	PUNCT
cana-5534	61	3	2025	2025	NUM
cana-5534	61	4	)	)	PUNCT
cana-5534	61	5	2576	2576	NUM
cana-5534	61	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-5534	61	7	theorem	theorem	VERB
cana-5534	61	8	1	1	NUM
cana-5534	61	9	(	(	PUNCT
cana-5534	61	10	multivariate	multivariate	NOUN
cana-5534	61	11	chernoff	chernoff	NOUN
cana-5534	61	12	bound	bind	VERB
cana-5534	61	13	)	)	PUNCT
cana-5534	61	14	.	.	PUNCT
cana-5534	62	1	let	let	VERB
cana-5534	62	2	𝑿	𝑿	PROPN
cana-5534	62	3	=	=	SYM
cana-5534	62	4	(	(	PUNCT
cana-5534	62	5	𝑋1	𝑋1	PROPN
cana-5534	62	6	,	,	PUNCT
cana-5534	62	7	𝑋2	𝑋2	VERB
cana-5534	62	8	,	,	PUNCT
cana-5534	62	9	.	.	PUNCT
cana-5534	62	10	.	.	PUNCT
cana-5534	63	1	.	.	PUNCT
cana-5534	64	1	,	,	PUNCT
cana-5534	64	2	𝑋𝑑)⊤	𝑋𝑑)⊤	VERB
cana-5534	64	3	∈	∈	PROPN
cana-5534	65	1	ℝ𝑑	ℝ𝑑	PROPN
cana-5534	65	2	be	be	AUX
cana-5534	65	3	a	a	DET
cana-5534	65	4	random	random	ADJ
cana-5534	65	5	vector	vector	NOUN
cana-5534	65	6	,	,	PUNCT
cana-5534	65	7	and	and	CCONJ
cana-5534	65	8	suppose	suppose	VERB
cana-5534	65	9	its	its	PRON
cana-5534	65	10	moment	moment	NOUN
cana-5534	65	11	generating	generate	VERB
cana-5534	65	12	function	function	NOUN
cana-5534	65	13	𝑀𝑿(𝝀	𝑀𝑿(𝝀	NOUN
cana-5534	65	14	)	)	PUNCT
cana-5534	66	1	=	=	SYM
cana-5534	66	2	𝐸	𝐸	PROPN
cana-5534	67	1	[	[	X
cana-5534	67	2	𝑒𝝀⊤𝑿	𝑒𝝀⊤𝑿	X
cana-5534	67	3	]	]	X
cana-5534	67	4	is	be	AUX
cana-5534	67	5	finite	finite	ADJ
cana-5534	67	6	for	for	ADP
cana-5534	67	7	all	all	DET
cana-5534	67	8	λ	λ	PROPN
cana-5534	67	9	∈	∈	PROPN
cana-5534	68	1	ℝ𝑑.	ℝ𝑑.	NOUN
cana-5534	68	2	then	then	ADV
cana-5534	68	3	,	,	PUNCT
cana-5534	68	4	for	for	ADP
cana-5534	68	5	any	any	DET
cana-5534	68	6	vector	vector	NOUN
cana-5534	68	7	a	a	DET
cana-5534	68	8	∈	∈	PROPN
cana-5534	68	9	ℝ𝑑	ℝ𝑑	PROPN
cana-5534	68	10	,	,	PUNCT
cana-5534	68	11	𝑃(𝑿	𝑃(𝑿	ADV
cana-5534	68	12	⪰	⪰	NOUN
cana-5534	68	13	𝒂	𝒂	NOUN
cana-5534	68	14	)	)	PUNCT
cana-5534	68	15	≤	≤	ADJ
cana-5534	68	16	𝑖𝑛𝑓𝝀≻𝟎	𝑖𝑛𝑓𝝀≻𝟎	PROPN
cana-5534	68	17	𝑀𝑿(𝝀	𝑀𝑿(𝝀	NOUN
cana-5534	68	18	)	)	PUNCT
cana-5534	68	19	𝑒𝝀⊤𝒂	𝑒𝝀⊤𝒂	VERB
cana-5534	68	20	where	where	SCONJ
cana-5534	68	21	x	x	PRON
cana-5534	68	22	⪰	⪰	VERB
cana-5534	68	23	a	a	DET
cana-5534	68	24	means	means	NOUN
cana-5534	68	25	𝑋𝑖	𝑋𝑖	PROPN
cana-5534	68	26	≥	≥	NOUN
cana-5534	68	27	𝑎𝑖	𝑎𝑖	ADP
cana-5534	68	28	for	for	ADP
cana-5534	68	29	all	all	DET
cana-5534	68	30	i	i	PRON
cana-5534	68	31	=	=	NOUN
cana-5534	68	32	1	1	NUM
cana-5534	68	33	,	,	PUNCT
cana-5534	68	34	.	.	PUNCT
cana-5534	68	35	.	.	PUNCT
cana-5534	68	36	.	.	PUNCT
cana-5534	69	1	,	,	PUNCT
cana-5534	69	2	d	d	X
cana-5534	69	3	,	,	PUNCT
cana-5534	69	4	and	and	CCONJ
cana-5534	69	5	λ	λ	X
cana-5534	69	6	≻	≻	PROPN
cana-5534	69	7	0	0	NUM
cana-5534	69	8	means	mean	VERB
cana-5534	69	9	𝜆𝑖	𝜆𝑖	PROPN
cana-5534	69	10	>	>	X
cana-5534	69	11	0	0	PUNCT
cana-5534	69	12	for	for	ADP
cana-5534	69	13	all	all	DET
cana-5534	69	14	i.	i.	NOUN
cana-5534	69	15	proof	proof	NOUN
cana-5534	69	16	.	.	PUNCT
cana-5534	70	1	let	let	VERB
cana-5534	70	2	λ	λ	X
cana-5534	70	3	∈	∈	PROPN
cana-5534	71	1	ℝ𝑑	ℝ𝑑	PROPN
cana-5534	71	2	be	be	AUX
cana-5534	71	3	such	such	ADJ
cana-5534	71	4	that	that	SCONJ
cana-5534	71	5	𝜆𝑖	𝜆𝑖	NOUN
cana-5534	71	6	>	>	X
cana-5534	71	7	0	0	PUNCT
cana-5534	71	8	for	for	SCONJ
cana-5534	71	9	all	all	DET
cana-5534	71	10	i.	i.	NOUN
cana-5534	71	11	consider	consider	VERB
cana-5534	71	12	the	the	DET
cana-5534	71	13	event	event	NOUN
cana-5534	71	14	x	x	PART
cana-5534	71	15	⪰	⪰	VERB
cana-5534	71	16	a	a	PRON
cana-5534	71	17	,	,	PUNCT
cana-5534	71	18	i.e.	i.e.	X
cana-5534	71	19	,	,	PUNCT
cana-5534	71	20	𝑋𝑖	𝑋𝑖	NOUN
cana-5534	71	21	≥	≥	NOUN
cana-5534	71	22	𝑎𝑖	𝑎𝑖	ADP
cana-5534	71	23	for	for	ADP
cana-5534	71	24	all	all	DET
cana-5534	71	25	i.	i.	NOUN
cana-5534	71	26	on	on	ADP
cana-5534	71	27	this	this	DET
cana-5534	71	28	event	event	NOUN
cana-5534	71	29	,	,	PUNCT
cana-5534	71	30	we	we	PRON
cana-5534	71	31	have	have	VERB
cana-5534	71	32	:	:	PUNCT
cana-5534	71	33	𝝀⊤𝑿	𝝀⊤𝑿	VERB
cana-5534	71	34	≥	≥	NUM
cana-5534	71	35	𝝀⊤𝒂.	𝝀⊤𝒂.	ADV
cana-5534	71	36	therefore	therefore	ADV
cana-5534	71	37	,	,	PUNCT
cana-5534	71	38	𝑃(𝑿	𝑃(𝑿	ADV
cana-5534	71	39	⪰	⪰	NOUN
cana-5534	71	40	𝒂	𝒂	NOUN
cana-5534	71	41	)	)	PUNCT
cana-5534	71	42	=	=	PUNCT
cana-5534	71	43	𝑃	𝑃	NOUN
cana-5534	71	44	(	(	PUNCT
cana-5534	71	45	𝑒𝝀⊤𝑿	𝑒𝝀⊤𝑿	NUM
cana-5534	71	46	≥	≥	NOUN
cana-5534	71	47	𝑒𝝀⊤𝒂	𝑒𝝀⊤𝒂	NUM
cana-5534	71	48	)	)	PUNCT
cana-5534	71	49	.	.	PUNCT
cana-5534	72	1	now	now	ADV
cana-5534	72	2	,	,	PUNCT
cana-5534	72	3	apply	apply	VERB
cana-5534	72	4	markov	markov	PROPN
cana-5534	72	5	’s	’s	PART
cana-5534	72	6	inequality	inequality	NOUN
cana-5534	72	7	to	to	ADP
cana-5534	72	8	the	the	DET
cana-5534	72	9	non	non	ADJ
cana-5534	72	10	-	-	ADJ
cana-5534	72	11	negative	negative	ADJ
cana-5534	72	12	random	random	ADJ
cana-5534	72	13	variable	variable	NOUN
cana-5534	72	14	𝑒𝝀⊤𝑿	𝑒𝝀⊤𝑿	NOUN
cana-5534	72	15	:	:	PUNCT
cana-5534	72	16	𝑃	𝑃	NOUN
cana-5534	72	17	(	(	PUNCT
cana-5534	72	18	𝑒𝝀⊤𝑿	𝑒𝝀⊤𝑿	NUM
cana-5534	72	19	≥	≥	NOUN
cana-5534	72	20	𝑒𝝀⊤𝒂	𝑒𝝀⊤𝒂	NUM
cana-5534	72	21	)	)	PUNCT
cana-5534	73	1	≤	≤	NUM
cana-5534	73	2	𝐸	𝐸	PROPN
cana-5534	73	3	[	[	X
cana-5534	73	4	𝑒𝝀⊤𝑿	𝑒𝝀⊤𝑿	NOUN
cana-5534	73	5	]	]	PUNCT
cana-5534	73	6	𝑒𝝀⊤𝒂	𝑒𝝀⊤𝒂	PROPN
cana-5534	73	7	=	=	SYM
cana-5534	73	8	𝑀𝑿(𝝀	𝑀𝑿(𝝀	NOUN
cana-5534	73	9	)	)	PUNCT
cana-5534	73	10	𝑒𝝀⊤𝒂	𝑒𝝀⊤𝒂	VERB
cana-5534	73	11	since	since	SCONJ
cana-5534	73	12	this	this	DET
cana-5534	73	13	inequality	inequality	NOUN
cana-5534	73	14	holds	hold	VERB
cana-5534	73	15	for	for	ADP
cana-5534	73	16	any	any	DET
cana-5534	73	17	λ	λ	PROPN
cana-5534	73	18	≻	≻	PROPN
cana-5534	73	19	0	0	NUM
cana-5534	73	20	,	,	PUNCT
cana-5534	73	21	we	we	PRON
cana-5534	73	22	can	can	AUX
cana-5534	73	23	minimize	minimize	VERB
cana-5534	73	24	the	the	DET
cana-5534	73	25	right	right	ADJ
cana-5534	73	26	-	-	PUNCT
cana-5534	73	27	hand	hand	NOUN
cana-5534	73	28	side	side	NOUN
cana-5534	73	29	to	to	PART
cana-5534	73	30	obtain	obtain	VERB
cana-5534	73	31	the	the	DET
cana-5534	73	32	tightest	tight	ADJ
cana-5534	73	33	bound	bind	VERB
cana-5534	73	34	:	:	PUNCT
cana-5534	73	35	𝑃(𝑿	𝑃(𝑿	ADV
cana-5534	73	36	⪰	⪰	NOUN
cana-5534	73	37	𝒂	𝒂	NOUN
cana-5534	73	38	)	)	PUNCT
cana-5534	73	39	≤	≤	NOUN
cana-5534	73	40	𝑖𝑛𝑓𝝀≻𝟎𝑀𝑿(𝝀	𝑖𝑛𝑓𝝀≻𝟎𝑀𝑿(𝝀	NUM
cana-5534	73	41	)	)	PUNCT
cana-5534	73	42	𝑒𝝀⊤𝒂	𝑒𝝀⊤𝒂	PROPN
cana-5534	73	43	.	.	PUNCT
cana-5534	74	1	this	this	PRON
cana-5534	74	2	completes	complete	VERB
cana-5534	74	3	the	the	DET
cana-5534	74	4	proof	proof	NOUN
cana-5534	74	5	.	.	PUNCT
cana-5534	75	1	this	this	DET
cana-5534	75	2	result	result	NOUN
cana-5534	75	3	is	be	AUX
cana-5534	75	4	particularly	particularly	ADV
cana-5534	75	5	useful	useful	ADJ
cana-5534	75	6	when	when	SCONJ
cana-5534	75	7	dealing	deal	VERB
cana-5534	75	8	with	with	ADP
cana-5534	75	9	rare	rare	ADJ
cana-5534	75	10	events	event	NOUN
cana-5534	75	11	in	in	ADP
cana-5534	75	12	high	high	ADJ
cana-5534	75	13	dimensions	dimension	NOUN
cana-5534	75	14	,	,	PUNCT
cana-5534	75	15	such	such	ADJ
cana-5534	75	16	as	as	ADP
cana-5534	75	17	the	the	DET
cana-5534	75	18	probability	probability	NOUN
cana-5534	75	19	that	that	SCONJ
cana-5534	75	20	multiple	multiple	ADJ
cana-5534	75	21	components	component	NOUN
cana-5534	75	22	of	of	ADP
cana-5534	75	23	a	a	DET
cana-5534	75	24	random	random	ADJ
cana-5534	75	25	vector	vector	NOUN
cana-5534	75	26	simultaneously	simultaneously	ADV
cana-5534	75	27	exceed	exceed	VERB
cana-5534	75	28	their	their	PRON
cana-5534	75	29	respective	respective	ADJ
cana-5534	75	30	thresholds	threshold	NOUN
cana-5534	75	31	.	.	PUNCT
cana-5534	76	1	direct	direct	ADJ
cana-5534	76	2	computation	computation	NOUN
cana-5534	76	3	of	of	ADP
cana-5534	76	4	such	such	ADJ
cana-5534	76	5	joint	joint	ADJ
cana-5534	76	6	tail	tail	NOUN
cana-5534	76	7	probabilities	probability	NOUN
cana-5534	76	8	is	be	AUX
cana-5534	76	9	often	often	ADV
cana-5534	76	10	intractable	intractable	ADJ
cana-5534	76	11	,	,	PUNCT
cana-5534	76	12	especially	especially	ADV
cana-5534	76	13	when	when	SCONJ
cana-5534	76	14	dependencies	dependency	NOUN
cana-5534	76	15	exist	exist	VERB
cana-5534	76	16	between	between	ADP
cana-5534	76	17	components	component	NOUN
cana-5534	76	18	.	.	PUNCT
cana-5534	77	1	the	the	DET
cana-5534	77	2	chernoff	chernoff	NOUN
cana-5534	77	3	bound	bind	VERB
cana-5534	77	4	circumvents	circumvent	VERB
cana-5534	77	5	this	this	PRON
cana-5534	77	6	by	by	ADP
cana-5534	77	7	converting	convert	VERB
cana-5534	77	8	the	the	DET
cana-5534	77	9	tail	tail	NOUN
cana-5534	77	10	probability	probability	NOUN
cana-5534	77	11	problem	problem	NOUN
cana-5534	77	12	into	into	ADP
cana-5534	77	13	an	an	DET
cana-5534	77	14	optimization	optimization	NOUN
cana-5534	77	15	over	over	ADP
cana-5534	77	16	exponential	exponential	ADJ
cana-5534	77	17	moments	moment	NOUN
cana-5534	77	18	the	the	DET
cana-5534	77	19	key	key	ADJ
cana-5534	77	20	idea	idea	NOUN
cana-5534	77	21	behind	behind	ADP
cana-5534	77	22	the	the	DET
cana-5534	77	23	chernoff	chernoff	NOUN
cana-5534	77	24	method	method	NOUN
cana-5534	77	25	is	be	AUX
cana-5534	77	26	based	base	VERB
cana-5534	77	27	on	on	ADP
cana-5534	77	28	markov	markov	PROPN
cana-5534	77	29	’s	’s	PART
cana-5534	77	30	inequality	inequality	NOUN
cana-5534	77	31	:	:	PUNCT
cana-5534	77	32	p(x	p(x	VERB
cana-5534	77	33	⪰	⪰	NOUN
cana-5534	77	34	a	a	PRON
cana-5534	77	35	)	)	PUNCT
cana-5534	78	1	=	=	SYM
cana-5534	78	2	p	p	X
cana-5534	78	3	(	(	PUNCT
cana-5534	78	4	𝑒𝝀⊤𝐗	𝑒𝝀⊤𝐗	X
cana-5534	78	5	≥	≥	NOUN
cana-5534	78	6	𝑒𝝀⊤𝒂	𝑒𝝀⊤𝒂	NOUN
cana-5534	78	7	)	)	PUNCT
cana-5534	78	8	≤	≤	NUM
cana-5534	78	9	𝐸	𝐸	PROPN
cana-5534	79	1	[	[	X
cana-5534	79	2	𝑒𝝀⊤𝑿	𝑒𝝀⊤𝑿	NOUN
cana-5534	79	3	]	]	PUNCT
cana-5534	79	4	𝑒𝝀⊤𝒂	𝑒𝝀⊤𝒂	X
cana-5534	79	5	,	,	PUNCT
cana-5534	79	6	for	for	ADP
cana-5534	79	7	any	any	DET
cana-5534	79	8	λ	λ	PROPN
cana-5534	79	9	≻	≻	PROPN
cana-5534	79	10	0	0	NUM
cana-5534	79	11	.	.	PUNCT
cana-5534	80	1	the	the	DET
cana-5534	80	2	tightest	tight	ADJ
cana-5534	80	3	bound	bind	VERB
cana-5534	80	4	is	be	AUX
cana-5534	80	5	then	then	ADV
cana-5534	80	6	obtained	obtain	VERB
cana-5534	80	7	by	by	ADP
cana-5534	80	8	minimizing	minimize	VERB
cana-5534	80	9	this	this	DET
cana-5534	80	10	ratio	ratio	NOUN
cana-5534	80	11	over	over	ADP
cana-5534	80	12	all	all	DET
cana-5534	80	13	such	such	ADJ
cana-5534	80	14	vectors	vector	NOUN
cana-5534	80	15	λ	λ	PROPN
cana-5534	80	16	.	.	PUNCT
cana-5534	81	1	the	the	DET
cana-5534	81	2	multivariate	multivariate	NOUN
cana-5534	81	3	chernoff	chernoff	NOUN
cana-5534	81	4	bound	bind	VERB
cana-5534	81	5	finds	find	VERB
cana-5534	81	6	use	use	NOUN
cana-5534	81	7	in	in	ADP
cana-5534	81	8	several	several	ADJ
cana-5534	81	9	area	area	NOUN
cana-5534	81	10	such	such	ADJ
cana-5534	81	11	as	as	ADP
cana-5534	81	12	:	:	PUNCT
cana-5534	81	13	finance	finance	NOUN
cana-5534	81	14	:	:	PUNCT
cana-5534	81	15	bounding	bound	VERB
cana-5534	81	16	joint	joint	ADJ
cana-5534	81	17	loss	loss	NOUN
cana-5534	81	18	probabilities	probability	NOUN
cana-5534	81	19	in	in	ADP
cana-5534	81	20	multi	multi	ADJ
cana-5534	81	21	-	-	ADJ
cana-5534	81	22	asset	asset	ADJ
cana-5534	81	23	portfolios	portfolio	NOUN
cana-5534	81	24	.	.	PUNCT
cana-5534	82	1	reliability	reliability	NOUN
cana-5534	82	2	:	:	PUNCT
cana-5534	82	3	evaluating	evaluate	VERB
cana-5534	82	4	failure	failure	NOUN
cana-5534	82	5	probabilities	probability	NOUN
cana-5534	82	6	in	in	ADP
cana-5534	82	7	redundant	redundant	ADJ
cana-5534	82	8	systems	system	NOUN
cana-5534	82	9	.	.	PUNCT
cana-5534	83	1	communications	communication	NOUN
cana-5534	83	2	on	on	ADP
cana-5534	83	3	applied	apply	VERB
cana-5534	83	4	nonlinear	nonlinear	ADJ
cana-5534	83	5	analysis	analysis	NOUN
cana-5534	83	6	issn	issn	NOUN
cana-5534	83	7	:	:	PUNCT
cana-5534	83	8	1074	1074	NUM
cana-5534	83	9	-	-	PUNCT
cana-5534	83	10	133x	133x	NUM
cana-5534	83	11	vol	vol	VERB
cana-5534	83	12	32	32	NUM
cana-5534	83	13	no	no	NOUN
cana-5534	83	14	.	.	PUNCT
cana-5534	84	1	10s	10	NOUN
cana-5534	84	2	(	(	PUNCT
cana-5534	84	3	2025	2025	NUM
cana-5534	84	4	)	)	PUNCT
cana-5534	84	5	2577	2577	NUM
cana-5534	84	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-5534	84	7	machine	machine	NOUN
cana-5534	84	8	learning	learning	NOUN
cana-5534	84	9	:	:	PUNCT
cana-5534	84	10	analyzing	analyze	VERB
cana-5534	84	11	the	the	DET
cana-5534	84	12	generalization	generalization	NOUN
cana-5534	84	13	error	error	NOUN
cana-5534	84	14	of	of	ADP
cana-5534	84	15	vector	vector	NOUN
cana-5534	84	16	-	-	PUNCT
cana-5534	84	17	valued	value	VERB
cana-5534	84	18	predictors	predictor	NOUN
cana-5534	84	19	.	.	PUNCT
cana-5534	85	1	information	information	NOUN
cana-5534	85	2	theory	theory	NOUN
cana-5534	85	3	:	:	PUNCT
cana-5534	85	4	bounding	bound	VERB
cana-5534	85	5	decoding	decode	VERB
cana-5534	85	6	error	error	NOUN
cana-5534	85	7	probabilities	probability	NOUN
cana-5534	85	8	for	for	ADP
cana-5534	85	9	vector	vector	NOUN
cana-5534	85	10	codes	code	NOUN
cana-5534	85	11	.	.	PUNCT
cana-5534	86	1	application	application	NOUN
cana-5534	86	2	:	:	PUNCT
cana-5534	86	3	portfolio	portfolio	NOUN
cana-5534	86	4	risk	risk	NOUN
cana-5534	86	5	assessment	assessment	NOUN
cana-5534	86	6	in	in	ADP
cana-5534	86	7	financial	financial	ADJ
cana-5534	86	8	risk	risk	NOUN
cana-5534	86	9	management	management	NOUN
cana-5534	86	10	,	,	PUNCT
cana-5534	86	11	the	the	DET
cana-5534	86	12	joint	joint	ADJ
cana-5534	86	13	behavior	behavior	NOUN
cana-5534	86	14	of	of	ADP
cana-5534	86	15	multiple	multiple	ADJ
cana-5534	86	16	asset	asset	NOUN
cana-5534	86	17	returns	return	NOUN
cana-5534	86	18	is	be	AUX
cana-5534	86	19	crucial	crucial	ADJ
cana-5534	86	20	for	for	ADP
cana-5534	86	21	understanding	understand	VERB
cana-5534	86	22	extreme	extreme	ADJ
cana-5534	86	23	market	market	NOUN
cana-5534	86	24	scenarios	scenario	NOUN
cana-5534	86	25	.	.	PUNCT
cana-5534	87	1	let	let	VERB
cana-5534	87	2	𝑿	𝑿	PROPN
cana-5534	87	3	=	=	SYM
cana-5534	87	4	(	(	PUNCT
cana-5534	87	5	𝑋1	𝑋1	PROPN
cana-5534	87	6	,	,	PUNCT
cana-5534	87	7	𝑋2	𝑋2	VERB
cana-5534	87	8	,	,	PUNCT
cana-5534	87	9	.	.	PUNCT
cana-5534	87	10	.	.	PUNCT
cana-5534	88	1	.	.	PUNCT
cana-5534	89	1	,	,	PUNCT
cana-5534	89	2	𝑋𝑑)⊤	𝑋𝑑)⊤	VERB
cana-5534	89	3	∈	∈	PROPN
cana-5534	90	1	ℝ𝑑	ℝ𝑑	PROPN
cana-5534	90	2	represent	represent	VERB
cana-5534	90	3	the	the	DET
cana-5534	90	4	random	random	ADJ
cana-5534	90	5	vector	vector	NOUN
cana-5534	90	6	of	of	ADP
cana-5534	90	7	returns	return	NOUN
cana-5534	90	8	for	for	ADP
cana-5534	90	9	d	d	PROPN
cana-5534	90	10	financial	financial	ADJ
cana-5534	90	11	assets	asset	NOUN
cana-5534	90	12	.	.	PUNCT
cana-5534	91	1	suppose	suppose	VERB
cana-5534	91	2	an	an	DET
cana-5534	91	3	investor	investor	NOUN
cana-5534	91	4	allocates	allocate	VERB
cana-5534	91	5	capital	capital	NOUN
cana-5534	91	6	according	accord	VERB
cana-5534	91	7	to	to	ADP
cana-5534	91	8	a	a	DET
cana-5534	91	9	weight	weight	NOUN
cana-5534	91	10	vector	vector	NOUN
cana-5534	91	11	w	w	PROPN
cana-5534	91	12	∈	∈	PROPN
cana-5534	92	1	ℝ𝑑	ℝ𝑑	PROPN
cana-5534	92	2	,	,	PUNCT
cana-5534	92	3	where	where	SCONJ
cana-5534	92	4	∑	∑	ADP
cana-5534	92	5	𝑤𝑖	𝑤𝑖	ADP
cana-5534	92	6	𝑑	𝑑	NOUN
cana-5534	92	7	𝑖=1	𝑖=1	PUNCT
cana-5534	92	8	=	=	SYM
cana-5534	92	9	1	1	NUM
cana-5534	92	10	𝑎𝑛𝑑	𝑎𝑛𝑑	ADJ
cana-5534	92	11	𝑤𝑖	𝑤𝑖	PRON
cana-5534	92	12	≥	≥	NOUN
cana-5534	92	13	0	0	NUM
cana-5534	92	14	the	the	DET
cana-5534	92	15	portfolio	portfolio	NOUN
cana-5534	92	16	return	return	NOUN
cana-5534	92	17	is	be	AUX
cana-5534	92	18	given	give	VERB
cana-5534	92	19	by	by	ADP
cana-5534	92	20	𝑅	𝑅	PROPN
cana-5534	92	21	=	=	PROPN
cana-5534	92	22	𝒘⊤𝑿.	𝒘⊤𝑿.	ADJ
cana-5534	92	23	in	in	ADP
cana-5534	92	24	risk	risk	NOUN
cana-5534	92	25	-	-	PUNCT
cana-5534	92	26	sensitive	sensitive	ADJ
cana-5534	92	27	applications	application	NOUN
cana-5534	92	28	such	such	ADJ
cana-5534	92	29	as	as	ADP
cana-5534	92	30	stress	stress	NOUN
cana-5534	92	31	testing	testing	NOUN
cana-5534	92	32	or	or	CCONJ
cana-5534	92	33	regulatory	regulatory	ADJ
cana-5534	92	34	compliance	compliance	NOUN
cana-5534	92	35	(	(	PUNCT
cana-5534	92	36	e.g.	e.g.	ADV
cana-5534	92	37	,	,	PUNCT
cana-5534	92	38	basel	basel	PROPN
cana-5534	92	39	iii	iii	PROPN
cana-5534	92	40	)	)	PUNCT
cana-5534	92	41	,	,	PUNCT
cana-5534	92	42	it	it	PRON
cana-5534	92	43	is	be	AUX
cana-5534	92	44	important	important	ADJ
cana-5534	92	45	to	to	PART
cana-5534	92	46	bound	bound	VERB
cana-5534	92	47	the	the	DET
cana-5534	92	48	probability	probability	NOUN
cana-5534	92	49	that	that	SCONJ
cana-5534	92	50	all	all	DET
cana-5534	92	51	asset	asset	NOUN
cana-5534	92	52	returns	return	NOUN
cana-5534	92	53	exceed	exceed	VERB
cana-5534	92	54	a	a	DET
cana-5534	92	55	specified	specify	VERB
cana-5534	92	56	threshold	threshold	NOUN
cana-5534	92	57	.	.	PUNCT
cana-5534	93	1	specifically	specifically	ADV
cana-5534	93	2	,	,	PUNCT
cana-5534	93	3	for	for	ADP
cana-5534	93	4	a	a	DET
cana-5534	93	5	threshold	threshold	NOUN
cana-5534	93	6	vector	vector	NOUN
cana-5534	93	7	a	a	DET
cana-5534	93	8	∈	∈	PROPN
cana-5534	94	1	ℝ𝑑	ℝ𝑑	PROPN
cana-5534	94	2	,	,	PUNCT
cana-5534	94	3	we	we	PRON
cana-5534	94	4	are	be	AUX
cana-5534	94	5	interested	interested	ADJ
cana-5534	94	6	in	in	ADP
cana-5534	94	7	estimating	estimate	VERB
cana-5534	94	8	the	the	DET
cana-5534	94	9	joint	joint	ADJ
cana-5534	94	10	tail	tail	NOUN
cana-5534	94	11	probability	probability	NOUN
cana-5534	94	12	p(x	p(x	NOUN
cana-5534	94	13	⪰	⪰	NOUN
cana-5534	94	14	a	a	PRON
cana-5534	94	15	)	)	PUNCT
cana-5534	94	16	=	=	PUNCT
cana-5534	94	17	p(𝑋1	p(𝑋1	PROPN
cana-5534	94	18	≥	≥	NOUN
cana-5534	94	19	𝑎1	𝑎1	NOUN
cana-5534	94	20	,	,	PUNCT
cana-5534	94	21	.	.	PUNCT
cana-5534	94	22	.	.	PUNCT
cana-5534	95	1	.	.	PUNCT
cana-5534	96	1	,	,	PUNCT
cana-5534	96	2	𝑋𝑑	𝑋𝑑	VERB
cana-5534	96	3	≥	≥	NOUN
cana-5534	96	4	𝑎d	𝑎d	VERB
cana-5534	96	5	)	)	PUNCT
cana-5534	96	6	.	.	PUNCT
cana-5534	97	1	assuming	assume	VERB
cana-5534	97	2	the	the	DET
cana-5534	97	3	moment	moment	NOUN
cana-5534	97	4	generating	generate	VERB
cana-5534	97	5	function	function	NOUN
cana-5534	97	6	mx(λ	mx(λ	PUNCT
cana-5534	97	7	)	)	PUNCT
cana-5534	97	8	=	=	SYM
cana-5534	97	9	e[𝑒𝝀⊤𝐗	e[𝑒𝝀⊤𝐗	PROPN
cana-5534	97	10	]	]	PUNCT
cana-5534	97	11	is	be	AUX
cana-5534	97	12	finite	finite	ADJ
cana-5534	97	13	for	for	ADP
cana-5534	97	14	λ	λ	PROPN
cana-5534	97	15	∈	∈	PROPN
cana-5534	97	16	ℝ𝑑	ℝ𝑑	PROPN
cana-5534	97	17	,	,	PUNCT
cana-5534	97	18	the	the	DET
cana-5534	97	19	multivariate	multivariate	NOUN
cana-5534	97	20	chernoff	chernoff	NOUN
cana-5534	97	21	bound	bind	VERB
cana-5534	97	22	provides	provide	VERB
cana-5534	97	23	the	the	DET
cana-5534	97	24	inequality	inequality	NOUN
cana-5534	97	25	:	:	PUNCT
cana-5534	97	26	𝑃(𝑿	𝑃(𝑿	ADV
cana-5534	97	27	⪰	⪰	NOUN
cana-5534	97	28	𝒂	𝒂	NOUN
cana-5534	97	29	)	)	PUNCT
cana-5534	97	30	≤	≤	ADJ
cana-5534	97	31	𝑖𝑛𝑓𝝀≻𝟎	𝑖𝑛𝑓𝝀≻𝟎	PROPN
cana-5534	97	32	𝑀𝑿(𝝀	𝑀𝑿(𝝀	NOUN
cana-5534	97	33	)	)	PUNCT
cana-5534	97	34	𝑒𝝀⊤𝒂	𝑒𝝀⊤𝒂	VERB
cana-5534	97	35	this	this	PRON
cana-5534	97	36	offers	offer	VERB
cana-5534	97	37	a	a	DET
cana-5534	97	38	conservative	conservative	ADJ
cana-5534	97	39	yet	yet	ADV
cana-5534	97	40	computationally	computationally	ADV
cana-5534	97	41	tractable	tractable	ADJ
cana-5534	97	42	upper	upper	ADJ
cana-5534	97	43	bound	bind	VERB
cana-5534	97	44	on	on	ADP
cana-5534	97	45	rare	rare	ADJ
cana-5534	97	46	-	-	PUNCT
cana-5534	97	47	event	event	NOUN
cana-5534	97	48	probabilities	probability	NOUN
cana-5534	97	49	.	.	PUNCT
cana-5534	98	1	numerical	numerical	ADJ
cana-5534	98	2	illustration	illustration	NOUN
cana-5534	98	3	:	:	PUNCT
cana-5534	98	4	bivariate	bivariate	ADJ
cana-5534	98	5	normal	normal	ADJ
cana-5534	98	6	portfolio	portfolio	NOUN
cana-5534	98	7	consider	consider	VERB
cana-5534	98	8	a	a	DET
cana-5534	98	9	portfolio	portfolio	NOUN
cana-5534	98	10	composed	compose	VERB
cana-5534	98	11	of	of	ADP
cana-5534	98	12	two	two	NUM
cana-5534	98	13	assets	asset	NOUN
cana-5534	98	14	,	,	PUNCT
cana-5534	98	15	where	where	SCONJ
cana-5534	98	16	the	the	DET
cana-5534	98	17	return	return	NOUN
cana-5534	98	18	vector	vector	NOUN
cana-5534	98	19	𝑿	𝑿	NOUN
cana-5534	98	20	=	=	SYM
cana-5534	98	21	(	(	PUNCT
cana-5534	98	22	𝑋1	𝑋1	PROPN
cana-5534	98	23	,	,	PUNCT
cana-5534	98	24	𝑋2)⊤	𝑋2)⊤	PROPN
cana-5534	98	25	follows	follow	VERB
cana-5534	98	26	a	a	DET
cana-5534	98	27	bivariate	bivariate	ADJ
cana-5534	98	28	normal	normal	ADJ
cana-5534	98	29	distribution	distribution	NOUN
cana-5534	98	30	:	:	PUNCT
cana-5534	98	31	𝑿	𝑿	ADJ
cana-5534	98	32	∼	∼	NOUN
cana-5534	98	33	𝑁2	𝑁2	NOUN
cana-5534	98	34	(	(	PUNCT
cana-5534	98	35	µ	µ	NOUN
cana-5534	98	36	=	=	PUNCT
cana-5534	98	37	(	(	PUNCT
cana-5534	98	38	𝟎.	𝟎.	X
cana-5534	98	39	𝟎𝟓	𝟎𝟓	NUM
cana-5534	98	40	𝟎.	𝟎.	NUM
cana-5534	98	41	𝟎𝟒	𝟎𝟒	NUM
cana-5534	98	42	)	)	PUNCT
cana-5534	98	43	,	,	PUNCT
cana-5534	98	44	𝛴	𝛴	PROPN
cana-5534	98	45	=	=	PUNCT
cana-5534	98	46	(	(	PUNCT
cana-5534	98	47	0.01	0.01	NUM
cana-5534	98	48	0.002	0.002	NUM
cana-5534	98	49	0.002	0.002	NUM
cana-5534	98	50	0.008	0.008	NUM
cana-5534	98	51	)	)	PUNCT
cana-5534	98	52	)	)	PUNCT
cana-5534	98	53	.	.	PUNCT
cana-5534	99	1	we	we	PRON
cana-5534	99	2	aim	aim	VERB
cana-5534	99	3	to	to	PART
cana-5534	99	4	compute	compute	VERB
cana-5534	99	5	an	an	DET
cana-5534	99	6	upper	upper	ADJ
cana-5534	99	7	bound	bind	VERB
cana-5534	99	8	on	on	ADP
cana-5534	99	9	the	the	DET
cana-5534	99	10	probability	probability	NOUN
cana-5534	99	11	:	:	PUNCT
cana-5534	99	12	p(𝑋1	p(𝑋1	PROPN
cana-5534	99	13	≥	≥	NUM
cana-5534	99	14	0.06	0.06	NUM
cana-5534	99	15	,	,	PUNCT
cana-5534	99	16	𝑋2	𝑋2	VERB
cana-5534	99	17	≥	≥	NOUN
cana-5534	99	18	0.05	0.05	NUM
cana-5534	99	19	)	)	PUNCT
cana-5534	99	20	,	,	PUNCT
cana-5534	99	21	using	use	VERB
cana-5534	99	22	the	the	DET
cana-5534	99	23	multivariate	multivariate	NOUN
cana-5534	99	24	chernoff	chernoff	NOUN
cana-5534	99	25	bound	bind	VERB
cana-5534	99	26	with	with	ADP
cana-5534	99	27	a	a	DET
cana-5534	99	28	trial	trial	NOUN
cana-5534	99	29	vector	vector	NOUN
cana-5534	99	30	𝝀	𝝀	NOUN
cana-5534	99	31	=	=	X
cana-5534	99	32	(	(	PUNCT
cana-5534	99	33	40	40	NUM
cana-5534	99	34	,	,	PUNCT
cana-5534	99	35	40)⊤.	40)⊤.	NUM
cana-5534	99	36	for	for	ADP
cana-5534	99	37	a	a	DET
cana-5534	99	38	multivariate	multivariate	NOUN
cana-5534	99	39	normal	normal	ADJ
cana-5534	99	40	distribution	distribution	NOUN
cana-5534	99	41	,	,	PUNCT
cana-5534	99	42	the	the	DET
cana-5534	99	43	moment	moment	NOUN
cana-5534	99	44	generating	generate	VERB
cana-5534	99	45	function	function	NOUN
cana-5534	99	46	is	be	AUX
cana-5534	99	47	:	:	PUNCT
cana-5534	99	48	𝑀𝑿(𝝀	𝑀𝑿(𝝀	NOUN
cana-5534	99	49	)	)	PUNCT
cana-5534	100	1	=	=	PRON
cana-5534	100	2	𝑒𝑥𝑝	𝑒𝑥𝑝	INTJ
cana-5534	100	3	(	(	PUNCT
cana-5534	100	4	𝝀⊤µ	𝝀⊤µ	PUNCT
cana-5534	101	1	+	+	NUM
cana-5534	101	2	1	1	NUM
cana-5534	101	3	2	2	NUM
cana-5534	101	4	𝝀⊤𝛴𝝀	𝝀⊤𝛴𝝀	NOUN
cana-5534	101	5	)	)	PUNCT
cana-5534	101	6	.	.	PUNCT
cana-5534	102	1	substituting	substitute	VERB
cana-5534	102	2	into	into	ADP
cana-5534	102	3	the	the	DET
cana-5534	102	4	bound	bind	VERB
cana-5534	102	5	,	,	PUNCT
cana-5534	102	6	we	we	PRON
cana-5534	102	7	have	have	VERB
cana-5534	102	8	:	:	PUNCT
cana-5534	102	9	𝑃(𝑿	𝑃(𝑿	ADV
cana-5534	102	10	⪰	⪰	NOUN
cana-5534	102	11	𝒂	𝒂	NOUN
cana-5534	102	12	)	)	PUNCT
cana-5534	102	13	≤	≤	NOUN
cana-5534	102	14	exp	exp	NOUN
cana-5534	102	15	(	(	PUNCT
cana-5534	102	16	𝝀⊤(µ	𝝀⊤(µ	INTJ
cana-5534	102	17	−	−	PROPN
cana-5534	102	18	𝒂	𝒂	X
cana-5534	102	19	)	)	PUNCT
cana-5534	102	20	+	+	CCONJ
cana-5534	102	21	1	1	NUM
cana-5534	102	22	2	2	NUM
cana-5534	102	23	𝝀⊤𝛴𝝀	𝝀⊤𝛴𝝀	NOUN
cana-5534	102	24	)	)	PUNCT
cana-5534	102	25	,	,	PUNCT
cana-5534	102	26	with	with	ADP
cana-5534	102	27	𝒂	𝒂	X
cana-5534	102	28	=	=	SYM
cana-5534	102	29	(	(	PUNCT
cana-5534	102	30	0.06	0.06	NUM
cana-5534	102	31	,	,	PUNCT
cana-5534	102	32	0.05)⊤.	0.05)⊤.	ADJ
cana-5534	102	33	computing	computing	NOUN
cana-5534	102	34	:	:	PUNCT
cana-5534	102	35	communications	communication	NOUN
cana-5534	102	36	on	on	ADP
cana-5534	102	37	applied	apply	VERB
cana-5534	102	38	nonlinear	nonlinear	ADJ
cana-5534	102	39	analysis	analysis	NOUN
cana-5534	102	40	issn	issn	NOUN
cana-5534	102	41	:	:	PUNCT
cana-5534	102	42	1074	1074	NUM
cana-5534	102	43	-	-	PUNCT
cana-5534	102	44	133x	133x	NUM
cana-5534	102	45	vol	vol	VERB
cana-5534	102	46	32	32	NUM
cana-5534	102	47	no	no	NOUN
cana-5534	102	48	.	.	PUNCT
cana-5534	103	1	10s	10	NOUN
cana-5534	103	2	(	(	PUNCT
cana-5534	103	3	2025	2025	NUM
cana-5534	103	4	)	)	PUNCT
cana-5534	103	5	2578	2578	NUM
cana-5534	103	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-5534	104	1	𝝀⊤(µ	𝝀⊤(µ	INTJ
cana-5534	104	2	−	−	PROPN
cana-5534	105	1	𝒂	𝒂	X
cana-5534	105	2	)	)	PUNCT
cana-5534	105	3	=	=	SYM
cana-5534	105	4	40(−0.01	40(−0.01	NOUN
cana-5534	105	5	)	)	PUNCT
cana-5534	106	1	+	+	NUM
cana-5534	106	2	40(−0.01	40(−0.01	NOUN
cana-5534	106	3	)	)	PUNCT
cana-5534	106	4	=	=	SYM
cana-5534	106	5	−0.8	−0.8	PROPN
cana-5534	106	6	,	,	PUNCT
cana-5534	106	7	𝝀⊤𝛴𝝀	𝝀⊤𝛴𝝀	NOUN
cana-5534	106	8	=	=	SYM
cana-5534	106	9	(	(	PUNCT
cana-5534	106	10	40	40	NUM
cana-5534	106	11	,	,	PUNCT
cana-5534	106	12	40	40	NUM
cana-5534	106	13	)	)	PUNCT
cana-5534	106	14	(	(	PUNCT
cana-5534	106	15	0.01	0.01	NUM
cana-5534	106	16	0.002	0.002	NUM
cana-5534	106	17	0.002	0.002	NUM
cana-5534	106	18	0.008	0.008	NUM
cana-5534	106	19	)	)	PUNCT
cana-5534	106	20	(	(	PUNCT
cana-5534	106	21	40	40	NUM
cana-5534	106	22	40	40	NUM
cana-5534	106	23	)	)	PUNCT
cana-5534	106	24	=	=	PUNCT
cana-5534	107	1	35.2	35.2	NUM
cana-5534	107	2	.	.	PUNCT
cana-5534	108	1	hence	hence	ADV
cana-5534	108	2	,	,	PUNCT
cana-5534	108	3	𝑃(𝑿	𝑃(𝑿	ADV
cana-5534	108	4	⪰	⪰	NOUN
cana-5534	108	5	𝒂	𝒂	NOUN
cana-5534	108	6	)	)	PUNCT
cana-5534	108	7	≤	≤	NOUN
cana-5534	108	8	𝑒𝑥𝑝(−0.8	𝑒𝑥𝑝(−0.8	NOUN
cana-5534	108	9	+	+	NOUN
cana-5534	108	10	17.6	17.6	NUM
cana-5534	108	11	)	)	PUNCT
cana-5534	108	12	=	=	SYM
cana-5534	108	13	𝑒𝑥𝑝(16.8	𝑒𝑥𝑝(16.8	NOUN
cana-5534	108	14	)	)	PUNCT
cana-5534	109	1	≈	≈	PROPN
cana-5534	109	2	1.95	1.95	NUM
cana-5534	109	3	×	×	NOUN
cana-5534	109	4	107	107	NUM
cana-5534	109	5	.	.	PUNCT
cana-5534	110	1	since	since	SCONJ
cana-5534	110	2	this	this	DET
cana-5534	110	3	bound	bind	VERB
cana-5534	110	4	exceeds	exceed	NOUN
cana-5534	110	5	1	1	NUM
cana-5534	110	6	,	,	PUNCT
cana-5534	110	7	it	it	PRON
cana-5534	110	8	is	be	AUX
cana-5534	110	9	uninformative	uninformative	ADJ
cana-5534	110	10	in	in	ADP
cana-5534	110	11	this	this	DET
cana-5534	110	12	case	case	NOUN
cana-5534	110	13	.	.	PUNCT
cana-5534	111	1	however	however	ADV
cana-5534	111	2	,	,	PUNCT
cana-5534	111	3	for	for	ADP
cana-5534	111	4	more	more	ADV
cana-5534	111	5	extreme	extreme	ADJ
cana-5534	111	6	thresholds	threshold	NOUN
cana-5534	111	7	(	(	PUNCT
cana-5534	111	8	e.g.	e.g.	ADV
cana-5534	111	9	,	,	PUNCT
cana-5534	111	10	𝒂	𝒂	X
cana-5534	111	11	=	=	PUNCT
cana-5534	111	12	(	(	PUNCT
cana-5534	111	13	0.08	0.08	NUM
cana-5534	111	14	,	,	PUNCT
cana-5534	111	15	0.07)⊤	0.07)⊤	NUM
cana-5534	111	16	)	)	PUNCT
cana-5534	111	17	,	,	PUNCT
cana-5534	111	18	the	the	DET
cana-5534	111	19	bound	bind	VERB
cana-5534	111	20	becomes	become	VERB
cana-5534	111	21	more	more	ADV
cana-5534	111	22	meaningful	meaningful	ADJ
cana-5534	111	23	:	:	PUNCT
cana-5534	111	24	𝝀⊤(µ	𝝀⊤(µ	PROPN
cana-5534	111	25	−	−	PROPN
cana-5534	112	1	𝒂	𝒂	X
cana-5534	112	2	)	)	PUNCT
cana-5534	112	3	=	=	NOUN
cana-5534	112	4	40(-0.03	40(-0.03	NUM
cana-5534	112	5	)	)	PUNCT
cana-5534	113	1	+	+	NUM
cana-5534	113	2	40(−0.03	40(−0.03	X
cana-5534	113	3	)	)	PUNCT
cana-5534	113	4	=	=	PUNCT
cana-5534	114	1	−2.4	−2.4	PROPN
cana-5534	114	2	⇒	⇒	VERB
cana-5534	114	3	𝑃(𝑿	𝑃(𝑿	ADV
cana-5534	114	4	⪰	⪰	NOUN
cana-5534	114	5	𝒂	𝒂	NOUN
cana-5534	114	6	)	)	PUNCT
cana-5534	114	7	≤	≤	NOUN
cana-5534	114	8	𝑒𝑥𝑝(15.2	𝑒𝑥𝑝(15.2	NOUN
cana-5534	114	9	)	)	PUNCT
cana-5534	115	1	≈	≈	PROPN
cana-5534	115	2	4.02	4.02	NUM
cana-5534	115	3	×	×	NOUN
cana-5534	115	4	106	106	NUM
cana-5534	115	5	.	.	PUNCT
cana-5534	116	1	the	the	DET
cana-5534	116	2	multivariate	multivariate	NOUN
cana-5534	116	3	chernoff	chernoff	NOUN
cana-5534	116	4	bound	bind	VERB
cana-5534	116	5	provides	provide	VERB
cana-5534	116	6	a	a	DET
cana-5534	116	7	tractable	tractable	ADJ
cana-5534	116	8	method	method	NOUN
cana-5534	116	9	to	to	PART
cana-5534	116	10	estimate	estimate	VERB
cana-5534	116	11	joint	joint	ADJ
cana-5534	116	12	tail	tail	NOUN
cana-5534	116	13	probabilities	probability	NOUN
cana-5534	116	14	.	.	PUNCT
cana-5534	117	1	although	although	SCONJ
cana-5534	117	2	the	the	DET
cana-5534	117	3	bound	bound	NOUN
cana-5534	117	4	may	may	AUX
cana-5534	117	5	be	be	AUX
cana-5534	117	6	loose	loose	ADJ
cana-5534	117	7	for	for	ADP
cana-5534	117	8	modest	modest	ADJ
cana-5534	117	9	deviations	deviation	NOUN
cana-5534	117	10	,	,	PUNCT
cana-5534	117	11	it	it	PRON
cana-5534	117	12	becomes	become	VERB
cana-5534	117	13	valuable	valuable	ADJ
cana-5534	117	14	in	in	ADP
cana-5534	117	15	stress	stress	NOUN
cana-5534	117	16	testing	test	VERB
cana-5534	117	17	scenarios	scenario	NOUN
cana-5534	117	18	where	where	SCONJ
cana-5534	117	19	evaluating	evaluate	VERB
cana-5534	117	20	the	the	DET
cana-5534	117	21	probability	probability	NOUN
cana-5534	117	22	of	of	ADP
cana-5534	117	23	rare	rare	ADJ
cana-5534	117	24	joint	joint	ADJ
cana-5534	117	25	exceedances	exceedance	NOUN
cana-5534	117	26	is	be	AUX
cana-5534	117	27	crucial	crucial	ADJ
cana-5534	117	28	.	.	PUNCT
cana-5534	118	1	example	example	NOUN
cana-5534	118	2	:	:	PUNCT
cana-5534	118	3	multivariate	multivariate	VERB
cana-5534	118	4	normal	normal	ADJ
cana-5534	118	5	distribution	distribution	NOUN
cana-5534	118	6	consider	consider	VERB
cana-5534	118	7	a	a	DET
cana-5534	118	8	random	random	ADJ
cana-5534	118	9	vector	vector	NOUN
cana-5534	118	10	𝑿	𝑿	NOUN
cana-5534	118	11	=	=	SYM
cana-5534	118	12	(	(	PUNCT
cana-5534	118	13	𝑋1	𝑋1	PROPN
cana-5534	118	14	,	,	PUNCT
cana-5534	118	15	𝑋2)⊤	𝑋2)⊤	ADP
cana-5534	118	16	following	follow	VERB
cana-5534	118	17	a	a	DET
cana-5534	118	18	multivariate	multivariate	NOUN
cana-5534	118	19	normal	normal	ADJ
cana-5534	118	20	distribution	distribution	NOUN
cana-5534	118	21	:	:	PUNCT
cana-5534	118	22	𝑿	𝑿	VERB
cana-5534	118	23	∼	∼	NOUN
cana-5534	118	24	𝑁(µ	𝑁(µ	NOUN
cana-5534	118	25	,	,	PUNCT
cana-5534	118	26	𝛴	𝛴	PROPN
cana-5534	118	27	)	)	PUNCT
cana-5534	118	28	,	,	PUNCT
cana-5534	118	29	where	where	SCONJ
cana-5534	118	30	the	the	DET
cana-5534	118	31	mean	mean	ADJ
cana-5534	118	32	vector	vector	NOUN
cana-5534	118	33	is	be	AUX
cana-5534	118	34	:	:	PUNCT
cana-5534	118	35	µ	µ	X
cana-5534	118	36	=	=	PUNCT
cana-5534	118	37	(	(	PUNCT
cana-5534	118	38	1	1	NUM
cana-5534	118	39	2	2	NUM
cana-5534	118	40	)	)	PUNCT
cana-5534	118	41	,	,	PUNCT
cana-5534	118	42	and	and	CCONJ
cana-5534	118	43	the	the	DET
cana-5534	118	44	covariance	covariance	NOUN
cana-5534	118	45	matrix	matrix	NOUN
cana-5534	118	46	is	be	AUX
cana-5534	118	47	:	:	PUNCT
cana-5534	118	48	𝛴	𝛴	PROPN
cana-5534	118	49	=	=	PUNCT
cana-5534	118	50	(	(	PUNCT
cana-5534	118	51	1	1	NUM
cana-5534	118	52	0.5	0.5	NUM
cana-5534	118	53	0.5	0.5	NUM
cana-5534	118	54	2	2	NUM
cana-5534	118	55	)	)	PUNCT
cana-5534	118	56	.	.	PUNCT
cana-5534	119	1	the	the	DET
cana-5534	119	2	moment	moment	NOUN
cana-5534	119	3	generating	generate	VERB
cana-5534	119	4	function	function	NOUN
cana-5534	119	5	(	(	PUNCT
cana-5534	119	6	mgf	mgf	PROPN
cana-5534	119	7	)	)	PUNCT
cana-5534	119	8	of	of	ADP
cana-5534	119	9	the	the	DET
cana-5534	119	10	multivariate	multivariate	NOUN
cana-5534	119	11	normal	normal	ADJ
cana-5534	119	12	distribution	distribution	NOUN
cana-5534	119	13	is	be	AUX
cana-5534	119	14	given	give	VERB
cana-5534	119	15	by	by	ADP
cana-5534	119	16	:	:	PUNCT
cana-5534	119	17	𝑀𝑿(𝝀	𝑀𝑿(𝝀	NOUN
cana-5534	119	18	)	)	PUNCT
cana-5534	120	1	=	=	PRON
cana-5534	120	2	𝑒𝑥𝑝	𝑒𝑥𝑝	INTJ
cana-5534	120	3	(	(	PUNCT
cana-5534	120	4	𝝀⊤µ	𝝀⊤µ	PUNCT
cana-5534	121	1	+	+	NUM
cana-5534	121	2	1	1	NUM
cana-5534	121	3	2	2	NUM
cana-5534	121	4	𝝀⊤𝛴𝝀	𝝀⊤𝛴𝝀	NOUN
cana-5534	121	5	)	)	PUNCT
cana-5534	121	6	,	,	PUNCT
cana-5534	121	7	where	where	SCONJ
cana-5534	121	8	𝝀	𝝀	NOUN
cana-5534	121	9	=	=	X
cana-5534	121	10	(	(	PUNCT
cana-5534	121	11	𝜆1	𝜆1	PROPN
cana-5534	121	12	,	,	PUNCT
cana-5534	121	13	𝜆2)⊤	𝜆2)⊤	PRON
cana-5534	121	14	is	be	AUX
cana-5534	121	15	a	a	DET
cana-5534	121	16	vector	vector	NOUN
cana-5534	121	17	of	of	ADP
cana-5534	121	18	parameters	parameter	NOUN
cana-5534	121	19	,	,	PUNCT
cana-5534	121	20	and	and	CCONJ
cana-5534	121	21	the	the	DET
cana-5534	121	22	expectation	expectation	NOUN
cana-5534	121	23	is	be	AUX
cana-5534	121	24	taken	take	VERB
cana-5534	121	25	over	over	ADP
cana-5534	121	26	the	the	DET
cana-5534	121	27	random	random	ADJ
cana-5534	121	28	vector	vector	NOUN
cana-5534	121	29	x.	x.	NOUN
cana-5534	122	1	we	we	PRON
cana-5534	122	2	are	be	AUX
cana-5534	122	3	interested	interested	ADJ
cana-5534	122	4	in	in	ADP
cana-5534	122	5	calculating	calculate	VERB
cana-5534	122	6	the	the	DET
cana-5534	122	7	upper	upper	ADJ
cana-5534	122	8	bound	bind	VERB
cana-5534	122	9	for	for	ADP
cana-5534	122	10	the	the	DET
cana-5534	122	11	probability	probability	NOUN
cana-5534	122	12	𝑃(𝑋1	𝑃(𝑋1	SYM
cana-5534	122	13	≥	≥	NUM
cana-5534	122	14	1.5	1.5	NUM
cana-5534	122	15	,	,	PUNCT
cana-5534	122	16	𝑋2	𝑋2	VERB
cana-5534	122	17	≥	≥	NOUN
cana-5534	122	18	2	2	NUM
cana-5534	122	19	)	)	PUNCT
cana-5534	122	20	using	use	VERB
cana-5534	122	21	the	the	DET
cana-5534	122	22	*	*	ADJ
cana-5534	122	23	*	*	NOUN
cana-5534	122	24	multivariate	multivariate	NOUN
cana-5534	122	25	chernoff	chernoff	NOUN
cana-5534	122	26	bound	bind	VERB
cana-5534	122	27	*	*	PROPN
cana-5534	122	28	*	*	PROPN
cana-5534	122	29	.	.	PUNCT
cana-5534	123	1	the	the	DET
cana-5534	123	2	chernoff	chernoff	NOUN
cana-5534	123	3	bound	bind	VERB
cana-5534	123	4	is	be	AUX
cana-5534	123	5	:	:	PUNCT
cana-5534	123	6	𝑃(𝑿	𝑃(𝑿	ADV
cana-5534	123	7	⪰	⪰	NOUN
cana-5534	123	8	𝒂	𝒂	NOUN
cana-5534	123	9	)	)	PUNCT
cana-5534	123	10	≤	≤	ADJ
cana-5534	123	11	𝑖𝑛𝑓𝝀≻𝟎	𝑖𝑛𝑓𝝀≻𝟎	PROPN
cana-5534	123	12	𝑀𝑿(𝝀	𝑀𝑿(𝝀	NOUN
cana-5534	123	13	)	)	PUNCT
cana-5534	123	14	𝑒𝝀⊤𝒂	𝑒𝝀⊤𝒂	PRON
cana-5534	123	15	,	,	PUNCT
cana-5534	123	16	where	where	SCONJ
cana-5534	123	17	𝒂	𝒂	X
cana-5534	123	18	=	=	X
cana-5534	123	19	(	(	PUNCT
cana-5534	123	20	1.5	1.5	NUM
cana-5534	123	21	2	2	NUM
cana-5534	123	22	)	)	PUNCT
cana-5534	123	23	is	be	AUX
cana-5534	123	24	the	the	DET
cana-5534	123	25	threshold	threshold	NOUN
cana-5534	123	26	vector	vector	NOUN
cana-5534	123	27	.	.	PUNCT
cana-5534	124	1	to	to	PART
cana-5534	124	2	apply	apply	VERB
cana-5534	124	3	the	the	DET
cana-5534	124	4	chernoff	chernoff	NOUN
cana-5534	124	5	bound	bind	VERB
cana-5534	124	6	,	,	PUNCT
cana-5534	124	7	we	we	PRON
cana-5534	124	8	need	need	VERB
cana-5534	124	9	to	to	PART
cana-5534	124	10	minimize	minimize	VERB
cana-5534	124	11	the	the	DET
cana-5534	124	12	following	follow	VERB
cana-5534	124	13	expression	expression	NOUN
cana-5534	124	14	:	:	PUNCT
cana-5534	124	15	𝑒𝑥𝑝	𝑒𝑥𝑝	INTJ
cana-5534	124	16	(	(	PUNCT
cana-5534	124	17	𝝀⊤µ	𝝀⊤µ	PUNCT
cana-5534	125	1	+	+	NUM
cana-5534	125	2	1	1	NUM
cana-5534	125	3	2	2	NUM
cana-5534	125	4	𝝀⊤𝛴𝝀	𝝀⊤𝛴𝝀	NOUN
cana-5534	125	5	)	)	PUNCT
cana-5534	125	6	𝑒𝝀⊤𝒂	𝑒𝝀⊤𝒂	PROPN
cana-5534	125	7	.	.	PUNCT
cana-5534	126	1	in	in	ADP
cana-5534	126	2	practice	practice	NOUN
cana-5534	126	3	,	,	PUNCT
cana-5534	126	4	this	this	DET
cana-5534	126	5	optimization	optimization	NOUN
cana-5534	126	6	is	be	AUX
cana-5534	126	7	typically	typically	ADV
cana-5534	126	8	solved	solve	VERB
cana-5534	126	9	using	use	VERB
cana-5534	126	10	numerical	numerical	ADJ
cana-5534	126	11	methods	method	NOUN
cana-5534	126	12	(	(	PUNCT
cana-5534	126	13	e.g.	e.g.	ADV
cana-5534	126	14	,	,	PUNCT
cana-5534	126	15	gradient	gradient	ADJ
cana-5534	126	16	descent	descent	NOUN
cana-5534	126	17	or	or	CCONJ
cana-5534	126	18	convex	convex	VERB
cana-5534	126	19	optimization	optimization	NOUN
cana-5534	126	20	techniques	technique	NOUN
cana-5534	126	21	)	)	PUNCT
cana-5534	126	22	.	.	PUNCT
cana-5534	127	1	communications	communication	NOUN
cana-5534	127	2	on	on	ADP
cana-5534	127	3	applied	apply	VERB
cana-5534	127	4	nonlinear	nonlinear	ADJ
cana-5534	127	5	analysis	analysis	NOUN
cana-5534	127	6	issn	issn	NOUN
cana-5534	127	7	:	:	PUNCT
cana-5534	127	8	1074	1074	NUM
cana-5534	127	9	-	-	PUNCT
cana-5534	127	10	133x	133x	NUM
cana-5534	127	11	vol	vol	VERB
cana-5534	127	12	32	32	NUM
cana-5534	127	13	no	no	NOUN
cana-5534	127	14	.	.	PUNCT
cana-5534	128	1	10s	10	NOUN
cana-5534	128	2	(	(	PUNCT
cana-5534	128	3	2025	2025	NUM
cana-5534	128	4	)	)	PUNCT
cana-5534	128	5	2579	2579	NUM
cana-5534	128	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-5534	128	7	particular	particular	ADJ
cana-5534	128	8	case	case	NOUN
cana-5534	128	9	.	.	PUNCT
cana-5534	129	1	let	let	VERB
cana-5534	129	2	us	we	PRON
cana-5534	129	3	choose	choose	VERB
cana-5534	129	4	λ	λ	X
cana-5534	129	5	=	=	SYM
cana-5534	129	6	(	(	PUNCT
cana-5534	129	7	1	1	NUM
cana-5534	129	8	,	,	PUNCT
cana-5534	129	9	1)⊤	1)⊤	NUM
cana-5534	129	10	and	and	CCONJ
cana-5534	129	11	calculate	calculate	VERB
cana-5534	129	12	the	the	DET
cana-5534	129	13	chernoff	chernoff	NOUN
cana-5534	129	14	bound	bind	VERB
cana-5534	129	15	.	.	PUNCT
cana-5534	130	1	first	first	ADV
cana-5534	130	2	,	,	PUNCT
cana-5534	130	3	we	we	PRON
cana-5534	130	4	compute	compute	VERB
cana-5534	130	5	the	the	DET
cana-5534	130	6	exponential	exponential	ADJ
cana-5534	130	7	part	part	NOUN
cana-5534	130	8	of	of	ADP
cana-5534	130	9	the	the	DET
cana-5534	130	10	chernoff	chernoff	NOUN
cana-5534	130	11	bound	bind	VERB
cana-5534	130	12	:	:	PUNCT
cana-5534	130	13	𝑒𝝀⊤𝒂	𝑒𝝀⊤𝒂	ADJ
cana-5534	130	14	=	=	NOUN
cana-5534	130	15	𝑒1.5	𝑒1.5	ADJ
cana-5534	130	16	+	+	ADJ
cana-5534	130	17	2	2	NUM
cana-5534	130	18	=	=	SYM
cana-5534	130	19	𝑒3.5	𝑒3.5	PROPN
cana-5534	130	20	≈	≈	PROPN
cana-5534	130	21	33.115	33.115	NUM
cana-5534	130	22	next	next	ADV
cana-5534	130	23	,	,	PUNCT
cana-5534	130	24	we	we	PRON
cana-5534	130	25	calculate	calculate	VERB
cana-5534	130	26	the	the	DET
cana-5534	130	27	mgf	mgf	PROPN
cana-5534	130	28	at	at	ADP
cana-5534	130	29	λ	λ	X
cana-5534	130	30	=	=	PUNCT
cana-5534	130	31	(	(	PUNCT
cana-5534	130	32	1	1	NUM
cana-5534	130	33	,	,	PUNCT
cana-5534	130	34	1)⊤	1)⊤	NUM
cana-5534	130	35	:	:	PUNCT
cana-5534	130	36	𝑀𝑿(1	𝑀𝑿(1	ADJ
cana-5534	130	37	,	,	PUNCT
cana-5534	130	38	1	1	NUM
cana-5534	130	39	)	)	PUNCT
cana-5534	130	40	=	=	PRON
cana-5534	130	41	𝑒𝑥𝑝	𝑒𝑥𝑝	INTJ
cana-5534	130	42	(	(	PUNCT
cana-5534	130	43	(	(	PUNCT
cana-5534	130	44	1	1	NUM
cana-5534	130	45	,	,	PUNCT
cana-5534	130	46	1)⊤	1)⊤	NUM
cana-5534	130	47	(	(	PUNCT
cana-5534	130	48	1	1	NUM
cana-5534	130	49	2	2	NUM
cana-5534	130	50	)	)	PUNCT
cana-5534	131	1	+	+	CCONJ
cana-5534	131	2	1	1	NUM
cana-5534	131	3	2	2	NUM
cana-5534	131	4	(	(	PUNCT
cana-5534	131	5	1	1	NUM
cana-5534	131	6	,	,	PUNCT
cana-5534	131	7	1)⊤	1)⊤	NUM
cana-5534	131	8	(	(	PUNCT
cana-5534	131	9	1	1	NUM
cana-5534	131	10	0.5	0.5	NUM
cana-5534	131	11	0.5	0.5	NUM
cana-5534	131	12	2	2	NUM
cana-5534	131	13	)	)	PUNCT
cana-5534	131	14	(	(	PUNCT
cana-5534	131	15	1	1	NUM
cana-5534	131	16	,	,	PUNCT
cana-5534	131	17	1)⊤	1)⊤	NUM
cana-5534	131	18	)	)	PUNCT
cana-5534	131	19	.	.	PUNCT
cana-5534	132	1	this	this	DET
cana-5534	132	2	simplifies	simplifie	NOUN
cana-5534	132	3	to	to	PART
cana-5534	132	4	:	:	PUNCT
cana-5534	132	5	𝑀𝑿(1	𝑀𝑿(1	PROPN
cana-5534	132	6	,	,	PUNCT
cana-5534	132	7	1	1	NUM
cana-5534	132	8	)	)	PUNCT
cana-5534	132	9	=	=	PRON
cana-5534	132	10	𝑒𝑥𝑝	𝑒𝑥𝑝	INTJ
cana-5534	132	11	(	(	PUNCT
cana-5534	132	12	1	1	NUM
cana-5534	132	13	+	+	NUM
cana-5534	132	14	2	2	NUM
cana-5534	132	15	+	+	CCONJ
cana-5534	132	16	1	1	NUM
cana-5534	132	17	2	2	NUM
cana-5534	132	18	(	(	PUNCT
cana-5534	132	19	1	1	NUM
cana-5534	132	20	+	+	NUM
cana-5534	132	21	2	2	NUM
cana-5534	132	22	+	+	NUM
cana-5534	132	23	0.5	0.5	NUM
cana-5534	132	24	+	+	NUM
cana-5534	132	25	0.5	0.5	NUM
cana-5534	132	26	)	)	PUNCT
cana-5534	132	27	)	)	PUNCT
cana-5534	133	1	=	=	PUNCT
cana-5534	133	2	𝑒𝑥𝑝	𝑒𝑥𝑝	INTJ
cana-5534	133	3	(	(	PUNCT
cana-5534	133	4	3	3	NUM
cana-5534	133	5	+	+	SYM
cana-5534	133	6	2	2	NUM
cana-5534	133	7	)	)	PUNCT
cana-5534	133	8	=	=	SYM
cana-5534	133	9	𝑒𝑥𝑝(5	𝑒𝑥𝑝(5	NOUN
cana-5534	133	10	)	)	PUNCT
cana-5534	133	11	.	.	PUNCT
cana-5534	134	1	thus	thus	ADV
cana-5534	134	2	,	,	PUNCT
cana-5534	134	3	the	the	DET
cana-5534	134	4	chernoff	chernoff	NOUN
cana-5534	134	5	bound	bind	VERB
cana-5534	134	6	is	be	AUX
cana-5534	134	7	:	:	PUNCT
cana-5534	134	8	𝑃(𝑋1	𝑃(𝑋1	NUM
cana-5534	134	9	≥	≥	NUM
cana-5534	134	10	1.5	1.5	NUM
cana-5534	134	11	,	,	PUNCT
cana-5534	134	12	𝑋2	𝑋2	VERB
cana-5534	134	13	≥	≥	NOUN
cana-5534	134	14	2	2	NUM
cana-5534	134	15	)	)	PUNCT
cana-5534	134	16	≤	≤	NOUN
cana-5534	134	17	𝑒𝑥𝑝(5	𝑒𝑥𝑝(5	NOUN
cana-5534	134	18	)	)	PUNCT
cana-5534	134	19	𝑒3.5	𝑒3.5	PROPN
cana-5534	134	20	=	=	SYM
cana-5534	134	21	𝑒𝑥𝑝(5	𝑒𝑥𝑝(5	NOUN
cana-5534	134	22	)	)	PUNCT
cana-5534	134	23	=	=	PUNCT
cana-5534	134	24	33.115	33.115	NUM
cana-5534	134	25	this	this	PRON
cana-5534	134	26	gives	give	VERB
cana-5534	134	27	us	we	PRON
cana-5534	134	28	the	the	DET
cana-5534	134	29	upper	upper	ADJ
cana-5534	134	30	bound	bind	VERB
cana-5534	134	31	on	on	ADP
cana-5534	134	32	the	the	DET
cana-5534	134	33	probability	probability	NOUN
cana-5534	134	34	.	.	PUNCT
cana-5534	135	1	in	in	ADP
cana-5534	135	2	practice	practice	NOUN
cana-5534	135	3	,	,	PUNCT
cana-5534	135	4	numerical	numerical	ADJ
cana-5534	135	5	optimization	optimization	NOUN
cana-5534	135	6	can	can	AUX
cana-5534	135	7	be	be	AUX
cana-5534	135	8	used	use	VERB
cana-5534	135	9	to	to	PART
cana-5534	135	10	obtain	obtain	VERB
cana-5534	135	11	tighter	tight	ADJ
cana-5534	135	12	bounds	bound	NOUN
cana-5534	135	13	by	by	ADP
cana-5534	135	14	adjusting	adjust	VERB
cana-5534	135	15	λ	λ	PROPN
cana-5534	135	16	.	.	PROPN
cana-5534	135	17	conclusion	conclusion	NOUN
cana-5534	135	18	and	and	CCONJ
cana-5534	135	19	perspectives	perspective	NOUN
cana-5534	135	20	in	in	ADP
cana-5534	135	21	this	this	DET
cana-5534	135	22	study	study	NOUN
cana-5534	135	23	,	,	PUNCT
cana-5534	135	24	we	we	PRON
cana-5534	135	25	reviewed	review	VERB
cana-5534	135	26	the	the	DET
cana-5534	135	27	generalized	generalized	ADJ
cana-5534	135	28	markov	markov	NOUN
cana-5534	135	29	inequality	inequality	NOUN
cana-5534	135	30	in	in	ADP
cana-5534	135	31	various	various	ADJ
cana-5534	135	32	ways	way	NOUN
cana-5534	135	33	,	,	PUNCT
cana-5534	135	34	showing	show	VERB
cana-5534	135	35	its	its	PRON
cana-5534	135	36	importance	importance	NOUN
cana-5534	135	37	in	in	ADP
cana-5534	135	38	probability	probability	NOUN
cana-5534	135	39	theory	theory	NOUN
cana-5534	135	40	.	.	PUNCT
cana-5534	136	1	by	by	ADP
cana-5534	136	2	looking	look	VERB
cana-5534	136	3	at	at	ADP
cana-5534	136	4	different	different	ADJ
cana-5534	136	5	convex	convex	NOUN
cana-5534	136	6	functions	function	NOUN
cana-5534	136	7	,	,	PUNCT
cana-5534	136	8	we	we	PRON
cana-5534	136	9	showed	show	VERB
cana-5534	136	10	that	that	SCONJ
cana-5534	136	11	the	the	DET
cana-5534	136	12	generalization	generalization	NOUN
cana-5534	136	13	can	can	AUX
cana-5534	136	14	provide	provide	VERB
cana-5534	136	15	tighter	tight	ADJ
cana-5534	136	16	bounds	bound	NOUN
cana-5534	136	17	than	than	ADP
cana-5534	136	18	the	the	DET
cana-5534	136	19	traditional	traditional	ADJ
cana-5534	136	20	form	form	NOUN
cana-5534	136	21	.	.	PUNCT
cana-5534	137	1	practical	practical	ADJ
cana-5534	137	2	examples	example	NOUN
cana-5534	137	3	confirmed	confirm	VERB
cana-5534	137	4	the	the	DET
cana-5534	137	5	usefulness	usefulness	NOUN
cana-5534	137	6	of	of	ADP
cana-5534	137	7	these	these	DET
cana-5534	137	8	bounds	bound	NOUN
cana-5534	137	9	,	,	PUNCT
cana-5534	137	10	particularly	particularly	ADV
cana-5534	137	11	regarding	regard	VERB
cana-5534	137	12	the	the	DET
cana-5534	137	13	tail	tail	NOUN
cana-5534	137	14	behavior	behavior	NOUN
cana-5534	137	15	of	of	ADP
cana-5534	137	16	distributions	distribution	NOUN
cana-5534	137	17	.	.	PUNCT
cana-5534	138	1	we	we	PRON
cana-5534	138	2	also	also	ADV
cana-5534	138	3	developed	develop	VERB
cana-5534	138	4	a	a	DET
cana-5534	138	5	multivariate	multivariate	NOUN
cana-5534	138	6	chernoff	chernoff	NOUN
cana-5534	138	7	bound	bind	VERB
cana-5534	138	8	to	to	PART
cana-5534	138	9	analyze	analyze	VERB
cana-5534	138	10	the	the	DET
cana-5534	138	11	joint	joint	ADJ
cana-5534	138	12	behavior	behavior	NOUN
cana-5534	138	13	of	of	ADP
cana-5534	138	14	random	random	ADJ
cana-5534	138	15	vectors	vector	NOUN
cana-5534	138	16	in	in	ADP
cana-5534	138	17	high	high	ADJ
cana-5534	138	18	-	-	PUNCT
cana-5534	138	19	dimensional	dimensional	ADJ
cana-5534	138	20	spaces	space	NOUN
cana-5534	138	21	.	.	PUNCT
cana-5534	139	1	this	this	PRON
cana-5534	139	2	has	have	VERB
cana-5534	139	3	applications	application	NOUN
cana-5534	139	4	in	in	ADP
cana-5534	139	5	areas	area	NOUN
cana-5534	139	6	like	like	ADP
cana-5534	139	7	risk	risk	NOUN
cana-5534	139	8	aggregation	aggregation	NOUN
cana-5534	139	9	and	and	CCONJ
cana-5534	139	10	machine	machine	NOUN
cana-5534	139	11	learning	learning	NOUN
cana-5534	139	12	.	.	PUNCT
cana-5534	140	1	future	future	ADJ
cana-5534	140	2	research	research	NOUN
cana-5534	140	3	could	could	AUX
cana-5534	140	4	explore	explore	VERB
cana-5534	140	5	using	use	VERB
cana-5534	140	6	data	datum	NOUN
cana-5534	140	7	-	-	PUNCT
cana-5534	140	8	dependent	dependent	ADJ
cana-5534	140	9	convex	convex	NOUN
cana-5534	140	10	functions	function	NOUN
cana-5534	140	11	for	for	ADP
cana-5534	140	12	optimized	optimize	VERB
cana-5534	140	13	bounds	bound	NOUN
cana-5534	140	14	,	,	PUNCT
cana-5534	140	15	implementing	implement	VERB
cana-5534	140	16	these	these	DET
cana-5534	140	17	bounds	bound	NOUN
cana-5534	140	18	in	in	ADP
cana-5534	140	19	learning	learn	VERB
cana-5534	140	20	algorithms	algorithm	NOUN
cana-5534	140	21	,	,	PUNCT
cana-5534	140	22	and	and	CCONJ
cana-5534	140	23	studying	study	VERB
cana-5534	140	24	generalized	generalized	ADJ
cana-5534	140	25	inequalities	inequality	NOUN
cana-5534	140	26	under	under	ADP
cana-5534	140	27	dependency	dependency	NOUN
cana-5534	140	28	structures	structure	NOUN
cana-5534	140	29	.	.	PUNCT
cana-5534	141	1	this	this	PRON
cana-5534	141	2	shows	show	VERB
cana-5534	141	3	that	that	SCONJ
cana-5534	141	4	the	the	DET
cana-5534	141	5	generalized	generalized	ADJ
cana-5534	141	6	markov	markov	NOUN
cana-5534	141	7	inequality	inequality	NOUN
cana-5534	141	8	is	be	AUX
cana-5534	141	9	an	an	DET
cana-5534	141	10	important	important	ADJ
cana-5534	141	11	tool	tool	NOUN
cana-5534	141	12	in	in	ADP
cana-5534	141	13	modern	modern	ADJ
cana-5534	141	14	probability	probability	NOUN
cana-5534	141	15	analysis	analysis	NOUN
cana-5534	141	16	.	.	PUNCT
cana-5534	142	1	references	reference	NOUN
cana-5534	142	2	[	[	X
cana-5534	142	3	1	1	X
cana-5534	142	4	]	]	PUNCT
cana-5534	142	5	s.	s.	PROPN
cana-5534	142	6	m.	m.	PROPN
cana-5534	142	7	ross	ross	PROPN
cana-5534	142	8	,	,	PUNCT
cana-5534	142	9	stochastic	stochastic	NOUN
cana-5534	142	10	processes	process	NOUN
cana-5534	142	11	,	,	PUNCT
cana-5534	142	12	john	john	PROPN
cana-5534	142	13	wiley	wiley	PROPN
cana-5534	142	14	&	&	CCONJ
cana-5534	142	15	sons	son	NOUN
cana-5534	142	16	,	,	PUNCT
cana-5534	142	17	2nd	2nd	PROPN
cana-5534	142	18	edition	edition	NOUN
cana-5534	142	19	,	,	PUNCT
cana-5534	142	20	1996	1996	NUM
cana-5534	142	21	.	.	PUNCT
cana-5534	143	1	[	[	X
cana-5534	143	2	2	2	NUM
cana-5534	143	3	]	]	PUNCT
cana-5534	143	4	s.	s.	PROPN
cana-5534	143	5	boucheron	boucheron	PROPN
cana-5534	143	6	,	,	PUNCT
cana-5534	143	7	g.	g.	PROPN
cana-5534	143	8	lugosi	lugosi	PROPN
cana-5534	143	9	,	,	PUNCT
cana-5534	143	10	and	and	CCONJ
cana-5534	143	11	p.	p.	PROPN
cana-5534	143	12	massart	massart	PROPN
cana-5534	143	13	,	,	PUNCT
cana-5534	143	14	concentration	concentration	NOUN
cana-5534	143	15	inequalities	inequality	NOUN
cana-5534	143	16	:	:	PUNCT
cana-5534	143	17	a	a	DET
cana-5534	143	18	nonasymptotic	nonasymptotic	ADJ
cana-5534	143	19	theory	theory	NOUN
cana-5534	143	20	of	of	ADP
cana-5534	143	21	independence	independence	NOUN
cana-5534	143	22	,	,	PUNCT
cana-5534	143	23	oxford	oxford	PROPN
cana-5534	143	24	university	university	PROPN
cana-5534	143	25	press	press	NOUN
cana-5534	143	26	,	,	PUNCT
cana-5534	143	27	2013	2013	NUM
cana-5534	143	28	.	.	PUNCT
cana-5534	144	1	[	[	X
cana-5534	144	2	3	3	NUM
cana-5534	144	3	]	]	PUNCT
cana-5534	144	4	a.	a.	NOUN
cana-5534	144	5	dembo	dembo	NOUN
cana-5534	144	6	and	and	CCONJ
cana-5534	144	7	o.	o.	PROPN
cana-5534	144	8	zeitouni	zeitouni	PROPN
cana-5534	144	9	,	,	PUNCT
cana-5534	144	10	large	large	ADJ
cana-5534	144	11	deviations	deviation	NOUN
cana-5534	144	12	techniques	technique	NOUN
cana-5534	144	13	and	and	CCONJ
cana-5534	144	14	applications	application	NOUN
cana-5534	144	15	,	,	PUNCT
cana-5534	144	16	springer	springer	NOUN
cana-5534	144	17	-	-	PUNCT
cana-5534	144	18	verlag	verlag	PROPN
cana-5534	144	19	,	,	PUNCT
cana-5534	144	20	2nd	2nd	PROPN
cana-5534	144	21	edition	edition	NOUN
cana-5534	144	22	,	,	PUNCT
cana-5534	144	23	1998	1998	NUM
cana-5534	144	24	.	.	PUNCT
cana-5534	145	1	[	[	X
cana-5534	145	2	4	4	X
cana-5534	145	3	]	]	PUNCT
cana-5534	145	4	t.	t.	NOUN
cana-5534	145	5	m.	m.	NOUN
cana-5534	145	6	cover	cover	NOUN
cana-5534	145	7	and	and	CCONJ
cana-5534	145	8	j.	j.	PROPN
cana-5534	145	9	a.	a.	PROPN
cana-5534	145	10	thomas	thomas	PROPN
cana-5534	145	11	,	,	PUNCT
cana-5534	145	12	elements	element	NOUN
cana-5534	145	13	of	of	ADP
cana-5534	145	14	information	information	NOUN
cana-5534	145	15	theory	theory	NOUN
cana-5534	145	16	,	,	PUNCT
cana-5534	145	17	wileyinterscience	wileyinterscience	NOUN
cana-5534	145	18	,	,	PUNCT
cana-5534	145	19	2nd	2nd	PROPN
cana-5534	145	20	edition	edition	NOUN
cana-5534	145	21	,	,	PUNCT
cana-5534	145	22	2006	2006	NUM
cana-5534	145	23	.	.	PUNCT
cana-5534	146	1	[	[	X
cana-5534	146	2	5	5	X
cana-5534	146	3	]	]	PUNCT
cana-5534	146	4	p.	p.	NOUN
cana-5534	146	5	embrechts	embrecht	NOUN
cana-5534	146	6	,	,	PUNCT
cana-5534	146	7	a.	a.	PROPN
cana-5534	146	8	j.	j.	PROPN
cana-5534	146	9	mcneil	mcneil	PROPN
cana-5534	146	10	,	,	PUNCT
cana-5534	146	11	and	and	CCONJ
cana-5534	146	12	d.	d.	PROPN
cana-5534	146	13	straumann	straumann	PROPN
cana-5534	146	14	,	,	PUNCT
cana-5534	146	15	quantitative	quantitative	ADJ
cana-5534	146	16	risk	risk	NOUN
cana-5534	146	17	management	management	NOUN
cana-5534	146	18	:	:	PUNCT
cana-5534	146	19	concepts	concept	NOUN
cana-5534	146	20	,	,	PUNCT
cana-5534	146	21	techniques	technique	NOUN
cana-5534	146	22	and	and	CCONJ
cana-5534	146	23	tools	tool	NOUN
cana-5534	146	24	,	,	PUNCT
cana-5534	146	25	princeton	princeton	PROPN
cana-5534	146	26	university	university	PROPN
cana-5534	146	27	press	press	NOUN
cana-5534	146	28	,	,	PUNCT
cana-5534	146	29	2005	2005	NUM
cana-5534	146	30	.	.	PUNCT