id	sid	tid	token	lemma	pos
ejpam-976	1	1	1_xxx_gupta.dvi	1_xxx_gupta.dvi	NUM
ejpam-976	1	2	european	european	ADJ
ejpam-976	1	3	journal	journal	NOUN
ejpam-976	1	4	of	of	ADP
ejpam-976	1	5	pure	pure	ADJ
ejpam-976	1	6	and	and	CCONJ
ejpam-976	1	7	applied	apply	VERB
ejpam-976	1	8	mathematics	mathematic	NOUN
ejpam-976	1	9	vol	vol	NOUN
ejpam-976	1	10	.	.	PUNCT
ejpam-976	2	1	3	3	NUM
ejpam-976	2	2	,	,	PUNCT
ejpam-976	2	3	no	no	INTJ
ejpam-976	2	4	.	.	NOUN
ejpam-976	2	5	5	5	NUM
ejpam-976	2	6	,	,	PUNCT
ejpam-976	2	7	2010	2010	NUM
ejpam-976	2	8	,	,	PUNCT
ejpam-976	2	9	779	779	NUM
ejpam-976	2	10	-	-	SYM
ejpam-976	2	11	785	785	NUM
ejpam-976	2	12	issn	issn	PROPN
ejpam-976	2	13	1307	1307	NUM
ejpam-976	2	14	-	-	SYM
ejpam-976	2	15	5543	5543	NUM
ejpam-976	2	16	–	–	PUNCT
ejpam-976	2	17	www.ejpam.com	www.ejpam.com	X
ejpam-976	2	18	convex	convex	NOUN
ejpam-976	2	19	ordering	ordering	NOUN
ejpam-976	2	20	of	of	ADP
ejpam-976	2	21	random	random	ADJ
ejpam-976	2	22	variables	variable	NOUN
ejpam-976	2	23	and	and	CCONJ
ejpam-976	2	24	its	its	PRON
ejpam-976	2	25	applications	application	NOUN
ejpam-976	2	26	in	in	ADP
ejpam-976	2	27	econometrics	econometrics	NOUN
ejpam-976	2	28	and	and	CCONJ
ejpam-976	2	29	actuarial	actuarial	ADJ
ejpam-976	2	30	science	science	NOUN
ejpam-976	2	31	arjun	arjun	PROPN
ejpam-976	2	32	k.	k.	PROPN
ejpam-976	2	33	gupta∗	gupta∗	PROPN
ejpam-976	2	34	,	,	PUNCT
ejpam-976	2	35	mohammad	mohammad	PROPN
ejpam-976	2	36	a.	a.	PROPN
ejpam-976	2	37	s.	s.	PROPN
ejpam-976	2	38	aziz	aziz	PROPN
ejpam-976	2	39	department	department	PROPN
ejpam-976	2	40	of	of	ADP
ejpam-976	2	41	mathematics	mathematics	PROPN
ejpam-976	2	42	and	and	CCONJ
ejpam-976	2	43	statistics	statistic	NOUN
ejpam-976	2	44	,	,	PUNCT
ejpam-976	2	45	bowling	bowl	VERB
ejpam-976	2	46	green	green	ADJ
ejpam-976	2	47	state	state	PROPN
ejpam-976	2	48	university	university	PROPN
ejpam-976	2	49	,	,	PUNCT
ejpam-976	2	50	bowling	bowling	NOUN
ejpam-976	2	51	green	green	NOUN
ejpam-976	2	52	,	,	PUNCT
ejpam-976	2	53	ohio	ohio	PROPN
ejpam-976	2	54	43403	43403	NUM
ejpam-976	2	55	,	,	PUNCT
ejpam-976	2	56	u.s.a	u.s.a	PROPN
ejpam-976	2	57	abstract	abstract	NOUN
ejpam-976	2	58	.	.	PUNCT
ejpam-976	3	1	it	it	PRON
ejpam-976	3	2	is	be	AUX
ejpam-976	3	3	well	well	ADV
ejpam-976	3	4	known	know	VERB
ejpam-976	3	5	that	that	SCONJ
ejpam-976	3	6	in	in	ADP
ejpam-976	3	7	economics	economic	NOUN
ejpam-976	3	8	and	and	CCONJ
ejpam-976	3	9	finance	finance	NOUN
ejpam-976	3	10	,	,	PUNCT
ejpam-976	3	11	the	the	DET
ejpam-976	3	12	data	datum	NOUN
ejpam-976	3	13	usually	usually	ADV
ejpam-976	3	14	have	have	VERB
ejpam-976	3	15	“	"	PUNCT
ejpam-976	3	16	fat	fat	ADJ
ejpam-976	3	17	tail	tail	NOUN
ejpam-976	3	18	”	"	PUNCT
ejpam-976	3	19	and	and	CCONJ
ejpam-976	3	20	in	in	ADP
ejpam-976	3	21	this	this	DET
ejpam-976	3	22	case	case	NOUN
ejpam-976	3	23	the	the	DET
ejpam-976	3	24	normal	normal	ADJ
ejpam-976	3	25	distribution	distribution	NOUN
ejpam-976	3	26	is	be	AUX
ejpam-976	3	27	not	not	PART
ejpam-976	3	28	a	a	DET
ejpam-976	3	29	good	good	ADJ
ejpam-976	3	30	model	model	NOUN
ejpam-976	3	31	to	to	PART
ejpam-976	3	32	use	use	VERB
ejpam-976	3	33	.	.	PUNCT
ejpam-976	4	1	the	the	DET
ejpam-976	4	2	skew	skew	ADJ
ejpam-976	4	3	normal	normal	ADJ
ejpam-976	4	4	distributions	distribution	NOUN
ejpam-976	4	5	recently	recently	ADV
ejpam-976	4	6	draw	draw	VERB
ejpam-976	4	7	considerable	considerable	ADJ
ejpam-976	4	8	attention	attention	NOUN
ejpam-976	4	9	as	as	ADP
ejpam-976	4	10	an	an	DET
ejpam-976	4	11	alternative	alternative	ADJ
ejpam-976	4	12	model	model	NOUN
ejpam-976	4	13	.	.	PUNCT
ejpam-976	5	1	unfortunately	unfortunately	ADV
ejpam-976	5	2	,	,	PUNCT
ejpam-976	5	3	the	the	DET
ejpam-976	5	4	distribution	distribution	NOUN
ejpam-976	5	5	of	of	ADP
ejpam-976	5	6	the	the	DET
ejpam-976	5	7	sum	sum	NOUN
ejpam-976	5	8	of	of	ADP
ejpam-976	5	9	log	log	NOUN
ejpam-976	5	10	-	-	PUNCT
ejpam-976	5	11	skew	skew	NOUN
ejpam-976	5	12	normal	normal	ADJ
ejpam-976	5	13	random	random	ADJ
ejpam-976	5	14	variables	variable	NOUN
ejpam-976	5	15	does	do	AUX
ejpam-976	5	16	not	not	PART
ejpam-976	5	17	have	have	VERB
ejpam-976	5	18	a	a	DET
ejpam-976	5	19	closed	closed	ADJ
ejpam-976	5	20	form	form	NOUN
ejpam-976	5	21	.	.	PUNCT
ejpam-976	6	1	in	in	ADP
ejpam-976	6	2	this	this	DET
ejpam-976	6	3	work	work	NOUN
ejpam-976	6	4	,	,	PUNCT
ejpam-976	6	5	we	we	PRON
ejpam-976	6	6	discuss	discuss	VERB
ejpam-976	6	7	the	the	DET
ejpam-976	6	8	use	use	NOUN
ejpam-976	6	9	of	of	ADP
ejpam-976	6	10	lower	low	ADJ
ejpam-976	6	11	convex	convex	ADJ
ejpam-976	6	12	order	order	NOUN
ejpam-976	6	13	of	of	ADP
ejpam-976	6	14	random	random	ADJ
ejpam-976	6	15	variables	variable	NOUN
ejpam-976	6	16	to	to	PART
ejpam-976	6	17	approximate	approximate	VERB
ejpam-976	6	18	this	this	DET
ejpam-976	6	19	distribution	distribution	NOUN
ejpam-976	6	20	.	.	PUNCT
ejpam-976	7	1	further	far	ADV
ejpam-976	7	2	,	,	PUNCT
ejpam-976	7	3	two	two	NUM
ejpam-976	7	4	application	application	NOUN
ejpam-976	7	5	of	of	ADP
ejpam-976	7	6	this	this	DET
ejpam-976	7	7	approximate	approximate	ADJ
ejpam-976	7	8	distribution	distribution	NOUN
ejpam-976	7	9	are	be	AUX
ejpam-976	7	10	given	give	VERB
ejpam-976	7	11	:	:	PUNCT
ejpam-976	7	12	first	first	ADV
ejpam-976	7	13	to	to	PART
ejpam-976	7	14	describe	describe	VERB
ejpam-976	7	15	the	the	DET
ejpam-976	7	16	final	final	ADJ
ejpam-976	7	17	wealth	wealth	NOUN
ejpam-976	7	18	of	of	ADP
ejpam-976	7	19	a	a	DET
ejpam-976	7	20	series	series	NOUN
ejpam-976	7	21	of	of	ADP
ejpam-976	7	22	payments	payment	NOUN
ejpam-976	7	23	,	,	PUNCT
ejpam-976	7	24	and	and	CCONJ
ejpam-976	7	25	second	second	ADV
ejpam-976	7	26	to	to	PART
ejpam-976	7	27	describe	describe	VERB
ejpam-976	7	28	the	the	DET
ejpam-976	7	29	present	present	ADJ
ejpam-976	7	30	value	value	NOUN
ejpam-976	7	31	of	of	ADP
ejpam-976	7	32	a	a	DET
ejpam-976	7	33	series	series	NOUN
ejpam-976	7	34	of	of	ADP
ejpam-976	7	35	payments	payment	NOUN
ejpam-976	7	36	.	.	PUNCT
ejpam-976	8	1	2000	2000	NUM
ejpam-976	8	2	mathematics	mathematic	NOUN
ejpam-976	8	3	subject	subject	NOUN
ejpam-976	8	4	classifications	classification	NOUN
ejpam-976	8	5	:	:	PUNCT
ejpam-976	8	6	primary	primary	ADJ
ejpam-976	8	7	62e17	62e17	NOUN
ejpam-976	8	8	;	;	PUNCT
ejpam-976	8	9	secondary	secondary	ADJ
ejpam-976	8	10	62p05	62p05	DET
ejpam-976	8	11	key	key	ADJ
ejpam-976	8	12	words	word	NOUN
ejpam-976	8	13	and	and	CCONJ
ejpam-976	8	14	phrases	phrase	NOUN
ejpam-976	8	15	:	:	PUNCT
ejpam-976	8	16	log	log	NOUN
ejpam-976	8	17	-	-	PUNCT
ejpam-976	8	18	skew	skew	NOUN
ejpam-976	8	19	normal	normal	ADJ
ejpam-976	8	20	random	random	ADJ
ejpam-976	8	21	variable	variable	NOUN
ejpam-976	8	22	,	,	PUNCT
ejpam-976	8	23	lower	low	ADJ
ejpam-976	8	24	convex	convex	ADJ
ejpam-976	8	25	order	order	NOUN
ejpam-976	8	26	bound	bind	VERB
ejpam-976	8	27	,	,	PUNCT
ejpam-976	8	28	comonotonocity	comonotonocity	NOUN
ejpam-976	8	29	1	1	NUM
ejpam-976	8	30	.	.	PUNCT
ejpam-976	8	31	preliminaries	preliminary	NOUN
ejpam-976	8	32	definition	definition	NOUN
ejpam-976	8	33	1	1	X
ejpam-976	8	34	.	.	PUNCT
ejpam-976	8	35	consider	consider	VERB
ejpam-976	8	36	two	two	NUM
ejpam-976	8	37	random	random	ADJ
ejpam-976	8	38	variables	variable	NOUN
ejpam-976	8	39	x	x	PUNCT
ejpam-976	8	40	and	and	CCONJ
ejpam-976	8	41	y	y	PRON
ejpam-976	8	42	such	such	ADJ
ejpam-976	8	43	that	that	SCONJ
ejpam-976	8	44	e[φ(x	e[φ(x	PROPN
ejpam-976	8	45	)	)	PUNCT
ejpam-976	8	46	]	]	PUNCT
ejpam-976	8	47	≤	≤	NUM
ejpam-976	8	48	e[φ(y	e[φ(y	NUM
ejpam-976	8	49	)	)	PUNCT
ejpam-976	8	50	]	]	PUNCT
ejpam-976	8	51	,	,	PUNCT
ejpam-976	8	52	for	for	ADP
ejpam-976	8	53	all	all	DET
ejpam-976	8	54	convex	convex	NOUN
ejpam-976	8	55	functions	function	NOUN
ejpam-976	8	56	φ	φ	NOUN
ejpam-976	8	57	,	,	PUNCT
ejpam-976	8	58	provided	provide	VERB
ejpam-976	8	59	expectation	expectation	NOUN
ejpam-976	8	60	exist.then	exist.then	ADP
ejpam-976	8	61	x	x	PRON
ejpam-976	8	62	is	be	AUX
ejpam-976	8	63	said	say	VERB
ejpam-976	8	64	to	to	PART
ejpam-976	8	65	be	be	AUX
ejpam-976	8	66	smaller	small	ADJ
ejpam-976	8	67	than	than	ADP
ejpam-976	8	68	y	y	PROPN
ejpam-976	8	69	in	in	ADP
ejpam-976	8	70	the	the	DET
ejpam-976	8	71	convex	convex	NOUN
ejpam-976	8	72	order	order	NOUN
ejpam-976	8	73	denoted	denote	VERB
ejpam-976	8	74	as	as	ADP
ejpam-976	8	75	x	x	X
ejpam-976	8	76	≤cx	≤cx	NOUN
ejpam-976	8	77	y	y	PROPN
ejpam-976	8	78	.	.	PUNCT
ejpam-976	9	1	definition	definition	NOUN
ejpam-976	9	2	2	2	NUM
ejpam-976	9	3	(	(	PUNCT
ejpam-976	9	4	convex	convex	NOUN
ejpam-976	9	5	order	order	NOUN
ejpam-976	9	6	definition	definition	NOUN
ejpam-976	9	7	using	use	VERB
ejpam-976	9	8	stop	stop	VERB
ejpam-976	9	9	-	-	PUNCT
ejpam-976	9	10	loss	loss	NOUN
ejpam-976	9	11	premium	premium	NOUN
ejpam-976	9	12	)	)	PUNCT
ejpam-976	9	13	.	.	PUNCT
ejpam-976	10	1	consider	consider	VERB
ejpam-976	10	2	two	two	NUM
ejpam-976	10	3	random	random	ADJ
ejpam-976	10	4	variables	variable	NOUN
ejpam-976	10	5	x	x	PUNCT
ejpam-976	10	6	and	and	CCONJ
ejpam-976	10	7	y.	y.	PROPN
ejpam-976	10	8	then	then	ADV
ejpam-976	10	9	x	x	PUNCT
ejpam-976	10	10	is	be	AUX
ejpam-976	10	11	said	say	VERB
ejpam-976	10	12	to	to	PART
ejpam-976	10	13	precede	precede	VERB
ejpam-976	10	14	y	y	PROPN
ejpam-976	10	15	in	in	ADP
ejpam-976	10	16	convex	convex	ADJ
ejpam-976	10	17	order	order	NOUN
ejpam-976	10	18	sense	sense	NOUN
ejpam-976	10	19	if	if	SCONJ
ejpam-976	10	20	and	and	CCONJ
ejpam-976	10	21	only	only	ADV
ejpam-976	10	22	if	if	SCONJ
ejpam-976	10	23	e[x	e[x	NOUN
ejpam-976	10	24	]	]	X
ejpam-976	11	1	=	=	PUNCT
ejpam-976	11	2	e[y	e[y	X
ejpam-976	11	3	]	]	PUNCT
ejpam-976	11	4	e[(x	e[(x	PROPN
ejpam-976	11	5	−	−	PROPN
ejpam-976	11	6	d)+]≤	d)+]≤	PROPN
ejpam-976	11	7	e[(y	e[(y	PROPN
ejpam-976	11	8	−	−	PROPN
ejpam-976	11	9	d)+	d)+	PROPN
ejpam-976	11	10	]	]	PUNCT
ejpam-976	11	11	,	,	PUNCT
ejpam-976	11	12	i(−∝,∝)(d	i(−∝,∝)(d	PROPN
ejpam-976	11	13	)	)	PUNCT
ejpam-976	12	1	where	where	SCONJ
ejpam-976	12	2	(	(	PUNCT
ejpam-976	12	3	x	x	SYM
ejpam-976	12	4	−	−	X
ejpam-976	12	5	d)+	d)+	NOUN
ejpam-976	12	6	=	=	SYM
ejpam-976	12	7	max(x	max(x	PROPN
ejpam-976	12	8	−	−	PROPN
ejpam-976	12	9	d	d	PROPN
ejpam-976	12	10	,	,	PUNCT
ejpam-976	12	11	0	0	NUM
ejpam-976	12	12	)	)	PUNCT
ejpam-976	12	13	∗corresponding	∗corresponde	VERB
ejpam-976	12	14	author	author	NOUN
ejpam-976	12	15	.	.	PUNCT
ejpam-976	13	1	email	email	NOUN
ejpam-976	13	2	address	address	NOUN
ejpam-976	13	3	:	:	PUNCT
ejpam-976	13	4	gupta�bgsu.edu	gupta�bgsu.edu	PROPN
ejpam-976	13	5	(	(	PUNCT
ejpam-976	13	6	arjun	arjun	PROPN
ejpam-976	13	7	k.	k.	PROPN
ejpam-976	13	8	gupta	gupta	PROPN
ejpam-976	13	9	)	)	PUNCT
ejpam-976	13	10	http://www.ejpam.com	http://www.ejpam.com	X
ejpam-976	14	1	779	779	NUM
ejpam-976	14	2	c	c	X
ejpam-976	14	3	©	©	PROPN
ejpam-976	14	4	2010	2010	NUM
ejpam-976	14	5	ejpam	ejpam	NOUN
ejpam-976	14	6	all	all	DET
ejpam-976	14	7	rights	right	NOUN
ejpam-976	14	8	reserved	reserve	VERB
ejpam-976	14	9	.	.	PUNCT
ejpam-976	15	1	a	a	DET
ejpam-976	15	2	gupta	gupta	PROPN
ejpam-976	15	3	,	,	PUNCT
ejpam-976	15	4	m	m	PROPN
ejpam-976	15	5	aziz	aziz	PROPN
ejpam-976	15	6	/	/	SYM
ejpam-976	15	7	eur	eur	PROPN
ejpam-976	15	8	.	.	PUNCT
ejpam-976	16	1	j.	j.	PROPN
ejpam-976	16	2	pure	pure	PROPN
ejpam-976	16	3	appl	appl	PROPN
ejpam-976	16	4	.	.	PROPN
ejpam-976	16	5	math	math	PROPN
ejpam-976	16	6	,	,	PUNCT
ejpam-976	16	7	3	3	NUM
ejpam-976	16	8	(	(	PUNCT
ejpam-976	16	9	2010	2010	NUM
ejpam-976	16	10	)	)	PUNCT
ejpam-976	16	11	,	,	PUNCT
ejpam-976	16	12	779	779	NUM
ejpam-976	16	13	-	-	SYM
ejpam-976	16	14	785	785	NUM
ejpam-976	16	15	780	780	NUM
ejpam-976	16	16	an	an	DET
ejpam-976	16	17	equivalent	equivalent	ADJ
ejpam-976	16	18	definition	definition	NOUN
ejpam-976	16	19	can	can	AUX
ejpam-976	16	20	be	be	AUX
ejpam-976	16	21	derived	derive	VERB
ejpam-976	16	22	from	from	ADP
ejpam-976	16	23	the	the	DET
ejpam-976	16	24	following	follow	VERB
ejpam-976	16	25	relation	relation	NOUN
ejpam-976	16	26	e[(x	e[(x	PROPN
ejpam-976	16	27	−	−	PROPN
ejpam-976	16	28	d)+]−	d)+]−	PROPN
ejpam-976	16	29	e[(d	e[(d	VERB
ejpam-976	16	30	−	−	NOUN
ejpam-976	16	31	x	x	SYM
ejpam-976	16	32	)	)	PUNCT
ejpam-976	17	1	+	+	ADP
ejpam-976	17	2	]	]	X
ejpam-976	17	3	=	=	X
ejpam-976	17	4	e(x	e(x	NOUN
ejpam-976	17	5	)	)	PUNCT
ejpam-976	18	1	−	−	PROPN
ejpam-976	18	2	d	d	PROPN
ejpam-976	18	3	for	for	ADP
ejpam-976	18	4	the	the	DET
ejpam-976	18	5	random	random	ADJ
ejpam-976	18	6	variable	variable	NOUN
ejpam-976	18	7	y	y	PROPN
ejpam-976	18	8	the	the	DET
ejpam-976	18	9	same	same	ADJ
ejpam-976	18	10	relation	relation	NOUN
ejpam-976	18	11	is	be	AUX
ejpam-976	18	12	given	give	VERB
ejpam-976	18	13	by	by	ADP
ejpam-976	18	14	,	,	PUNCT
ejpam-976	18	15	e[(y	e[(y	PROPN
ejpam-976	18	16	−	−	PROPN
ejpam-976	19	1	d)+]−	d)+]−	PROPN
ejpam-976	19	2	e[(d	e[(d	VERB
ejpam-976	19	3	−	−	PROPN
ejpam-976	19	4	y	y	PROPN
ejpam-976	19	5	)	)	PUNCT
ejpam-976	20	1	+	+	ADP
ejpam-976	20	2	]	]	X
ejpam-976	20	3	=	=	PUNCT
ejpam-976	20	4	e(y	e(y	ADJ
ejpam-976	20	5	)	)	PUNCT
ejpam-976	20	6	−	−	PROPN
ejpam-976	21	1	d	d	X
ejpam-976	21	2	now	now	ADV
ejpam-976	21	3	assume	assume	VERB
ejpam-976	21	4	x	x	PUNCT
ejpam-976	21	5	≤cx	≤cx	PROPN
ejpam-976	21	6	y	y	PROPN
ejpam-976	21	7	,	,	PUNCT
ejpam-976	21	8	which	which	PRON
ejpam-976	21	9	implies	imply	VERB
ejpam-976	21	10	that	that	SCONJ
ejpam-976	21	11	e[x	e[x	NOUN
ejpam-976	21	12	]	]	X
ejpam-976	21	13	=	=	PUNCT
ejpam-976	21	14	e[y	e[y	X
ejpam-976	21	15	]	]	PUNCT
ejpam-976	21	16	and	and	CCONJ
ejpam-976	21	17	e[(x	e[(x	PROPN
ejpam-976	21	18	−	−	PROPN
ejpam-976	21	19	d)+]≤	d)+]≤	PROPN
ejpam-976	21	20	e[(y	e[(y	PROPN
ejpam-976	21	21	−	−	PROPN
ejpam-976	21	22	d)+	d)+	PROPN
ejpam-976	21	23	]	]	PUNCT
ejpam-976	21	24	,	,	PUNCT
ejpam-976	21	25	i(−∝,∝)(d	i(−∝,∝)(d	PROPN
ejpam-976	21	26	)	)	PUNCT
ejpam-976	21	27	hence	hence	ADV
ejpam-976	21	28	e[(d	e[(d	VERB
ejpam-976	21	29	−	−	NOUN
ejpam-976	21	30	x	x	SYM
ejpam-976	21	31	)	)	PUNCT
ejpam-976	22	1	+	+	ADP
ejpam-976	22	2	]	]	X
ejpam-976	22	3	≤	≤	NUM
ejpam-976	22	4	e[(d	e[(d	VERB
ejpam-976	22	5	−	−	PROPN
ejpam-976	22	6	y	y	PROPN
ejpam-976	22	7	)	)	PUNCT
ejpam-976	23	1	+	+	ADP
ejpam-976	23	2	]	]	X
ejpam-976	23	3	therefore	therefore	ADV
ejpam-976	23	4	,	,	PUNCT
ejpam-976	23	5	a	a	DET
ejpam-976	23	6	definition	definition	NOUN
ejpam-976	23	7	equivalent	equivalent	ADJ
ejpam-976	23	8	to	to	ADP
ejpam-976	23	9	the	the	DET
ejpam-976	23	10	definition	definition	NOUN
ejpam-976	23	11	here	here	ADV
ejpam-976	23	12	is	be	AUX
ejpam-976	23	13	e[x	e[x	NOUN
ejpam-976	23	14	]	]	PUNCT
ejpam-976	23	15	=	=	PUNCT
ejpam-976	23	16	e[y	e[y	NOUN
ejpam-976	23	17	]	]	PUNCT
ejpam-976	23	18	e[(d	e[(d	VERB
ejpam-976	23	19	−	−	NOUN
ejpam-976	23	20	x	x	SYM
ejpam-976	23	21	)	)	PUNCT
ejpam-976	24	1	+	+	ADP
ejpam-976	24	2	]	]	X
ejpam-976	24	3	≤	≤	NUM
ejpam-976	24	4	e[(d	e[(d	VERB
ejpam-976	24	5	−	−	PROPN
ejpam-976	24	6	y	y	PROPN
ejpam-976	24	7	)	)	PUNCT
ejpam-976	25	1	+	+	ADP
ejpam-976	25	2	]	]	X
ejpam-976	25	3	1.1	1.1	NUM
ejpam-976	25	4	.	.	PUNCT
ejpam-976	25	5	properties	property	NOUN
ejpam-976	25	6	of	of	ADP
ejpam-976	25	7	convex	convex	NOUN
ejpam-976	25	8	ordering	ordering	NOUN
ejpam-976	25	9	of	of	ADP
ejpam-976	25	10	random	random	ADJ
ejpam-976	25	11	variables	variable	NOUN
ejpam-976	25	12	1	1	NUM
ejpam-976	25	13	.	.	PUNCT
ejpam-976	26	1	if	if	SCONJ
ejpam-976	26	2	x	x	PRON
ejpam-976	26	3	precedes	precede	VERB
ejpam-976	26	4	y	y	PROPN
ejpam-976	26	5	in	in	ADP
ejpam-976	26	6	convex	convex	ADJ
ejpam-976	26	7	order	order	NOUN
ejpam-976	26	8	sense	sense	NOUN
ejpam-976	26	9	i.e	i.e	INTJ
ejpam-976	26	10	if	if	SCONJ
ejpam-976	26	11	x	x	PRON
ejpam-976	26	12	≤cx	≤cx	NOUN
ejpam-976	26	13	y	y	PROPN
ejpam-976	26	14	,	,	PUNCT
ejpam-976	26	15	then	then	ADV
ejpam-976	26	16	e[x	e[x	X
ejpam-976	26	17	]	]	X
ejpam-976	27	1	=	=	PUNCT
ejpam-976	27	2	e[y	e[y	X
ejpam-976	27	3	]	]	PUNCT
ejpam-976	27	4	and	and	CCONJ
ejpam-976	27	5	var[x	var[x	PROPN
ejpam-976	27	6	]	]	PUNCT
ejpam-976	27	7	≤	≤	NUM
ejpam-976	27	8	var[y	var[y	NOUN
ejpam-976	27	9	]	]	X
ejpam-976	28	1	2	2	X
ejpam-976	28	2	.	.	PUNCT
ejpam-976	28	3	if	if	SCONJ
ejpam-976	28	4	x	x	PRON
ejpam-976	28	5	≤cx	≤cx	VERB
ejpam-976	28	6	y	y	PROPN
ejpam-976	28	7	and	and	CCONJ
ejpam-976	28	8	z	z	PROPN
ejpam-976	28	9	is	be	AUX
ejpam-976	28	10	independent	independent	ADJ
ejpam-976	28	11	of	of	ADP
ejpam-976	28	12	x	x	X
ejpam-976	28	13	and	and	CCONJ
ejpam-976	28	14	y	y	PROPN
ejpam-976	28	15	then	then	ADV
ejpam-976	28	16	x	x	X
ejpam-976	29	1	+	+	CCONJ
ejpam-976	29	2	z	z	NOUN
ejpam-976	29	3	≤cx	≤cx	NOUN
ejpam-976	29	4	y	y	PROPN
ejpam-976	30	1	+	+	CCONJ
ejpam-976	30	2	z	z	NOUN
ejpam-976	30	3	3	3	X
ejpam-976	30	4	.	.	PUNCT
ejpam-976	31	1	let	let	VERB
ejpam-976	31	2	x	x	PRON
ejpam-976	31	3	and	and	CCONJ
ejpam-976	31	4	y	y	PROPN
ejpam-976	31	5	be	be	AUX
ejpam-976	31	6	two	two	NUM
ejpam-976	31	7	random	random	ADJ
ejpam-976	31	8	variables	variable	NOUN
ejpam-976	31	9	,	,	PUNCT
ejpam-976	31	10	then	then	ADV
ejpam-976	31	11	x	x	PART
ejpam-976	31	12	≤cx	≤cx	NOUN
ejpam-976	31	13	y	y	PROPN
ejpam-976	31	14	⇔−x	⇔−x	X
ejpam-976	31	15	≤cx	≤cx	ADJ
ejpam-976	31	16	−y	−y	VERB
ejpam-976	31	17	4	4	NUM
ejpam-976	31	18	.	.	PUNCT
ejpam-976	32	1	let	let	VERB
ejpam-976	32	2	x	x	PRON
ejpam-976	32	3	and	and	CCONJ
ejpam-976	32	4	y	y	PROPN
ejpam-976	32	5	be	be	AUX
ejpam-976	32	6	two	two	NUM
ejpam-976	32	7	random	random	ADJ
ejpam-976	32	8	variables	variable	NOUN
ejpam-976	32	9	such	such	ADJ
ejpam-976	32	10	that	that	DET
ejpam-976	32	11	e[x	e[x	NOUN
ejpam-976	32	12	]	]	X
ejpam-976	32	13	=	=	PUNCT
ejpam-976	32	14	e[y	e[y	VERB
ejpam-976	32	15	]	]	PUNCT
ejpam-976	32	16	.	.	PUNCT
ejpam-976	33	1	then	then	ADV
ejpam-976	33	2	x	x	PUNCT
ejpam-976	33	3	≤cx	≤cx	NOUN
ejpam-976	33	4	y	y	PROPN
ejpam-976	33	5	if	if	SCONJ
ejpam-976	33	6	and	and	CCONJ
ejpam-976	33	7	only	only	ADV
ejpam-976	33	8	if	if	SCONJ
ejpam-976	33	9	e|x	e|x	PROPN
ejpam-976	33	10	−	−	PROPN
ejpam-976	33	11	a|	a|	PROPN
ejpam-976	33	12	≤cx	≤cx	NOUN
ejpam-976	33	13	e|y	e|y	VERB
ejpam-976	33	14	−	−	PROPN
ejpam-976	34	1	a|,∀	a|,∀	NOUN
ejpam-976	34	2	a	a	DET
ejpam-976	34	3	∈	∈	PROPN
ejpam-976	34	4	ℜ	ℜ	ADJ
ejpam-976	34	5	5	5	NUM
ejpam-976	34	6	.	.	PUNCT
ejpam-976	35	1	the	the	DET
ejpam-976	35	2	convex	convex	ADJ
ejpam-976	35	3	order	order	NOUN
ejpam-976	35	4	is	be	AUX
ejpam-976	35	5	closed	close	VERB
ejpam-976	35	6	under	under	ADP
ejpam-976	35	7	mixtures	mixture	NOUN
ejpam-976	35	8	:	:	PUNCT
ejpam-976	35	9	let	let	VERB
ejpam-976	35	10	x	x	PRON
ejpam-976	35	11	,	,	PUNCT
ejpam-976	35	12	y	y	PROPN
ejpam-976	35	13	and	and	CCONJ
ejpam-976	35	14	θ	θ	PROPN
ejpam-976	35	15	be	be	VERB
ejpam-976	35	16	random	random	ADJ
ejpam-976	35	17	variables	variable	NOUN
ejpam-976	35	18	such	such	ADJ
ejpam-976	35	19	that	that	SCONJ
ejpam-976	35	20	[	[	X
ejpam-976	35	21	x	x	X
ejpam-976	35	22	|θ	|θ	NOUN
ejpam-976	35	23	=	=	SYM
ejpam-976	35	24	θ	θ	NOUN
ejpam-976	35	25	]	]	X
ejpam-976	35	26	≤cx	≤cx	NOUN
ejpam-976	36	1	[	[	X
ejpam-976	36	2	y	y	PROPN
ejpam-976	36	3	|θ=	|θ=	NOUN
ejpam-976	36	4	θ]∀θ	θ]∀θ	NOUN
ejpam-976	36	5	in	in	ADP
ejpam-976	36	6	the	the	DET
ejpam-976	36	7	support	support	NOUN
ejpam-976	36	8	of	of	ADP
ejpam-976	36	9	θ	θ	PROPN
ejpam-976	36	10	.	.	PUNCT
ejpam-976	37	1	then	then	ADV
ejpam-976	37	2	x	x	SYM
ejpam-976	37	3	≤cx	≤cx	NOUN
ejpam-976	37	4	y	y	PROPN
ejpam-976	37	5	.	.	PUNCT
ejpam-976	38	1	6	6	X
ejpam-976	38	2	.	.	PUNCT
ejpam-976	38	3	the	the	DET
ejpam-976	38	4	convex	convex	ADJ
ejpam-976	38	5	order	order	NOUN
ejpam-976	38	6	is	be	AUX
ejpam-976	38	7	closed	close	VERB
ejpam-976	38	8	under	under	ADP
ejpam-976	38	9	convolution	convolution	NOUN
ejpam-976	38	10	:	:	PUNCT
ejpam-976	38	11	let	let	VERB
ejpam-976	38	12	x1	x1	NUM
ejpam-976	38	13	,	,	PUNCT
ejpam-976	38	14	x2	x2	PROPN
ejpam-976	38	15	,	,	PUNCT
ejpam-976	38	16	.	.	PUNCT
ejpam-976	38	17	.	.	PUNCT
ejpam-976	39	1	.	.	PUNCT
ejpam-976	40	1	,	,	PUNCT
ejpam-976	40	2	xm	xm	PROPN
ejpam-976	40	3	be	be	AUX
ejpam-976	40	4	a	a	DET
ejpam-976	40	5	set	set	NOUN
ejpam-976	40	6	of	of	ADP
ejpam-976	40	7	independent	independent	ADJ
ejpam-976	40	8	random	random	ADJ
ejpam-976	40	9	variables	variable	NOUN
ejpam-976	40	10	and	and	CCONJ
ejpam-976	40	11	y1	y1	NOUN
ejpam-976	40	12	,	,	PUNCT
ejpam-976	40	13	y2	y2	INTJ
ejpam-976	40	14	,	,	PUNCT
ejpam-976	40	15	.	.	PUNCT
ejpam-976	40	16	.	.	PUNCT
ejpam-976	41	1	.	.	PUNCT
ejpam-976	42	1	,	,	PUNCT
ejpam-976	42	2	yn	yn	PRON
ejpam-976	42	3	be	be	VERB
ejpam-976	42	4	another	another	DET
ejpam-976	42	5	set	set	NOUN
ejpam-976	42	6	of	of	ADP
ejpam-976	42	7	independent	independent	ADJ
ejpam-976	42	8	random	random	ADJ
ejpam-976	42	9	variables	variable	NOUN
ejpam-976	42	10	.	.	PUNCT
ejpam-976	43	1	if	if	SCONJ
ejpam-976	43	2	x	x	PRON
ejpam-976	43	3	i	i	PRON
ejpam-976	43	4	≤cx	≤cx	VERB
ejpam-976	43	5	yi	yi	PROPN
ejpam-976	43	6	,	,	PUNCT
ejpam-976	43	7	for	for	ADP
ejpam-976	43	8	i	i	PROPN
ejpam-976	43	9	=	=	NOUN
ejpam-976	43	10	1	1	NUM
ejpam-976	43	11	,	,	PUNCT
ejpam-976	43	12	....	....	PUNCT
ejpam-976	43	13	,	,	PUNCT
ejpam-976	43	14	m	m	PROPN
ejpam-976	43	15	,	,	PUNCT
ejpam-976	43	16	then	then	ADV
ejpam-976	43	17	∑m	∑m	PROPN
ejpam-976	43	18	j=1	j=1	NOUN
ejpam-976	44	1	x	x	PUNCT
ejpam-976	44	2	j	j	PROPN
ejpam-976	44	3	≤cx	≤cx	NOUN
ejpam-976	44	4	∑m	∑m	PROPN
ejpam-976	44	5	j=1	j=1	PROPN
ejpam-976	44	6	yj	yj	PROPN
ejpam-976	44	7	7	7	PROPN
ejpam-976	44	8	.	.	PUNCT
ejpam-976	45	1	let	let	VERB
ejpam-976	45	2	x	x	PRON
ejpam-976	45	3	be	be	AUX
ejpam-976	45	4	a	a	DET
ejpam-976	45	5	random	random	ADJ
ejpam-976	45	6	variable	variable	NOUN
ejpam-976	45	7	with	with	ADP
ejpam-976	45	8	finite	finite	ADJ
ejpam-976	45	9	mean	mean	NOUN
ejpam-976	45	10	.	.	PUNCT
ejpam-976	46	1	then	then	ADV
ejpam-976	46	2	x	x	X
ejpam-976	47	1	+	+	PUNCT
ejpam-976	47	2	e[x	e[x	NOUN
ejpam-976	47	3	]	]	PUNCT
ejpam-976	47	4	≤cx	≤cx	NOUN
ejpam-976	47	5	2x	2x	NUM
ejpam-976	47	6	8	8	NUM
ejpam-976	47	7	.	.	PUNCT
ejpam-976	48	1	let	let	VERB
ejpam-976	48	2	x1	x1	NUM
ejpam-976	48	3	,	,	PUNCT
ejpam-976	48	4	x2	x2	PROPN
ejpam-976	48	5	,	,	PUNCT
ejpam-976	48	6	.	.	PUNCT
ejpam-976	48	7	.	.	PUNCT
ejpam-976	49	1	.	.	PUNCT
ejpam-976	50	1	,	,	PUNCT
ejpam-976	50	2	xm	xm	PROPN
ejpam-976	50	3	and	and	CCONJ
ejpam-976	50	4	y	y	PROPN
ejpam-976	50	5	be	be	AUX
ejpam-976	50	6	(	(	PUNCT
ejpam-976	50	7	n+1	n+1	NOUN
ejpam-976	50	8	)	)	PUNCT
ejpam-976	50	9	random	random	ADJ
ejpam-976	50	10	variables	variable	NOUN
ejpam-976	50	11	.	.	PUNCT
ejpam-976	51	1	if	if	SCONJ
ejpam-976	51	2	x	x	PRON
ejpam-976	51	3	i	i	PRON
ejpam-976	51	4	≤cx	≤cx	VERB
ejpam-976	51	5	y	y	PROPN
ejpam-976	51	6	,	,	PUNCT
ejpam-976	51	7	i	i	PRON
ejpam-976	51	8	=	=	NOUN
ejpam-976	51	9	1	1	NUM
ejpam-976	51	10	,	,	PUNCT
ejpam-976	51	11	....	....	PUNCT
ejpam-976	51	12	,	,	PUNCT
ejpam-976	51	13	n	n	CCONJ
ejpam-976	51	14	,	,	PUNCT
ejpam-976	51	15	then∑n	then∑n	ADP
ejpam-976	51	16	i=1	i=1	PROPN
ejpam-976	51	17	aix	aix	PROPN
ejpam-976	51	18	i	i	PRON
ejpam-976	51	19	≤cx	≤cx	VERB
ejpam-976	51	20	y	y	PROPN
ejpam-976	51	21	,	,	PUNCT
ejpam-976	51	22	whenever	whenever	SCONJ
ejpam-976	51	23	ai	ai	VERB
ejpam-976	51	24	≥	≥	NOUN
ejpam-976	51	25	0,i	0,i	NUM
ejpam-976	52	1	=	=	SYM
ejpam-976	52	2	1	1	NUM
ejpam-976	52	3	,	,	PUNCT
ejpam-976	52	4	.	.	PUNCT
ejpam-976	52	5	.	.	PUNCT
ejpam-976	53	1	.	.	PUNCT
ejpam-976	54	1	,	,	PUNCT
ejpam-976	54	2	n	n	PROPN
ejpam-976	54	3	and	and	CCONJ
ejpam-976	54	4	∑n	∑n	PROPN
ejpam-976	54	5	i=1	i=1	PROPN
ejpam-976	54	6	ai	ai	VERB
ejpam-976	54	7	=	=	ADJ
ejpam-976	54	8	1	1	NUM
ejpam-976	54	9	a	a	DET
ejpam-976	54	10	gupta	gupta	PROPN
ejpam-976	54	11	,	,	PUNCT
ejpam-976	54	12	m	m	PROPN
ejpam-976	54	13	aziz	aziz	PROPN
ejpam-976	54	14	/	/	SYM
ejpam-976	54	15	eur	eur	PROPN
ejpam-976	54	16	.	.	PUNCT
ejpam-976	55	1	j.	j.	PROPN
ejpam-976	55	2	pure	pure	PROPN
ejpam-976	55	3	appl	appl	PROPN
ejpam-976	55	4	.	.	PROPN
ejpam-976	55	5	math	math	PROPN
ejpam-976	55	6	,	,	PUNCT
ejpam-976	55	7	3	3	NUM
ejpam-976	55	8	(	(	PUNCT
ejpam-976	55	9	2010	2010	NUM
ejpam-976	55	10	)	)	PUNCT
ejpam-976	55	11	,	,	PUNCT
ejpam-976	55	12	779	779	NUM
ejpam-976	55	13	-	-	SYM
ejpam-976	55	14	785	785	NUM
ejpam-976	55	15	781	781	NUM
ejpam-976	55	16	9	9	NUM
ejpam-976	55	17	.	.	PUNCT
ejpam-976	56	1	let	let	VERB
ejpam-976	56	2	x	x	PRON
ejpam-976	56	3	and	and	CCONJ
ejpam-976	56	4	y	y	PROPN
ejpam-976	56	5	be	be	AUX
ejpam-976	56	6	independent	independent	ADJ
ejpam-976	56	7	random	random	ADJ
ejpam-976	56	8	variables	variable	NOUN
ejpam-976	56	9	.	.	PUNCT
ejpam-976	57	1	then	then	ADV
ejpam-976	57	2	x	x	X
ejpam-976	57	3	i	i	PRON
ejpam-976	57	4	≤cx	≤cx	VERB
ejpam-976	57	5	yi	yi	PROPN
ejpam-976	58	1	if	if	SCONJ
ejpam-976	58	2	and	and	CCONJ
ejpam-976	58	3	only	only	ADV
ejpam-976	58	4	if	if	SCONJ
ejpam-976	58	5	e[φ(x	e[φ(x	PROPN
ejpam-976	58	6	,	,	PUNCT
ejpam-976	58	7	y	y	PROPN
ejpam-976	58	8	)	)	PUNCT
ejpam-976	58	9	]	]	PUNCT
ejpam-976	58	10	≤	≤	X
ejpam-976	58	11	e[φ(y	e[φ(y	NUM
ejpam-976	58	12	,	,	PUNCT
ejpam-976	58	13	x	x	X
ejpam-976	58	14	)	)	PUNCT
ejpam-976	58	15	]	]	PUNCT
ejpam-976	58	16	∀φ	∀φ	X
ejpam-976	58	17	∈	∈	NOUN
ejpam-976	58	18	℘cx	℘cx	NOUN
ejpam-976	58	19	,	,	PUNCT
ejpam-976	58	20	where	where	SCONJ
ejpam-976	58	21	℘cx	℘cx	NOUN
ejpam-976	58	22	=	=	PRON
ejpam-976	58	23	{	{	PUNCT
ejpam-976	58	24	φ	φ	NOUN
ejpam-976	58	25	:	:	PUNCT
ejpam-976	58	26	ℜ2	ℜ2	VERB
ejpam-976	58	27	−→ℜ	−→ℜ	X
ejpam-976	58	28	:	:	PUNCT
ejpam-976	58	29	φ(x	φ(x	PROPN
ejpam-976	58	30	,	,	PUNCT
ejpam-976	58	31	y	y	PROPN
ejpam-976	58	32	)	)	PUNCT
ejpam-976	58	33	−φ(y	−φ(y	VERB
ejpam-976	58	34	,	,	PUNCT
ejpam-976	58	35	x	x	PUNCT
ejpam-976	58	36	)	)	PUNCT
ejpam-976	58	37	is	be	AUX
ejpam-976	58	38	convex	convex	ADJ
ejpam-976	58	39	for	for	ADP
ejpam-976	58	40	all	all	DET
ejpam-976	58	41	x	x	SYM
ejpam-976	58	42	∈	∈	PROPN
ejpam-976	58	43	y	y	PROPN
ejpam-976	58	44	}	}	PUNCT
ejpam-976	58	45	.	.	PUNCT
ejpam-976	59	1	10	10	NUM
ejpam-976	59	2	.	.	PUNCT
ejpam-976	60	1	let	let	VERB
ejpam-976	60	2	x1	x1	PROPN
ejpam-976	60	3	and	and	CCONJ
ejpam-976	60	4	x2	x2	PROPN
ejpam-976	60	5	be	be	VERB
ejpam-976	60	6	a	a	DET
ejpam-976	60	7	pair	pair	NOUN
ejpam-976	60	8	of	of	ADP
ejpam-976	60	9	independent	independent	ADJ
ejpam-976	60	10	random	random	ADJ
ejpam-976	60	11	variables	variable	NOUN
ejpam-976	60	12	and	and	CCONJ
ejpam-976	60	13	let	let	VERB
ejpam-976	60	14	y1	y1	VERB
ejpam-976	60	15	and	and	CCONJ
ejpam-976	60	16	y2	y2	PROPN
ejpam-976	60	17	be	be	AUX
ejpam-976	60	18	another	another	DET
ejpam-976	60	19	pair	pair	NOUN
ejpam-976	60	20	of	of	ADP
ejpam-976	60	21	independent	independent	ADJ
ejpam-976	60	22	random	random	ADJ
ejpam-976	60	23	variables	variable	NOUN
ejpam-976	60	24	.	.	PUNCT
ejpam-976	61	1	if	if	SCONJ
ejpam-976	61	2	x	x	PRON
ejpam-976	61	3	i	i	PRON
ejpam-976	61	4	≤cx	≤cx	VERB
ejpam-976	61	5	yi	yi	PROPN
ejpam-976	61	6	,	,	PUNCT
ejpam-976	61	7	i	i	PRON
ejpam-976	61	8	=	=	NOUN
ejpam-976	61	9	1,2	1,2	NUM
ejpam-976	61	10	then	then	ADV
ejpam-976	61	11	x1x2	x1x2	PUNCT
ejpam-976	61	12	≤cx	≤cx	PROPN
ejpam-976	61	13	y1y2	y1y2	PROPN
ejpam-976	61	14	.	.	PROPN
ejpam-976	61	15	2	2	NUM
ejpam-976	61	16	.	.	X
ejpam-976	61	17	main	main	ADJ
ejpam-976	61	18	result	result	NOUN
ejpam-976	61	19	of	of	ADP
ejpam-976	61	20	convex	convex	PROPN
ejpam-976	61	21	ordering	ordering	NOUN
ejpam-976	61	22	theorem	theorem	NOUN
ejpam-976	61	23	1	1	NUM
ejpam-976	61	24	.	.	X
ejpam-976	62	1	for	for	ADP
ejpam-976	62	2	any	any	DET
ejpam-976	62	3	random	random	ADJ
ejpam-976	62	4	vector	vector	NOUN
ejpam-976	62	5	x	x	NOUN
ejpam-976	62	6	=	=	SYM
ejpam-976	62	7	(	(	PUNCT
ejpam-976	62	8	x1	x1	PROPN
ejpam-976	62	9	,	,	PUNCT
ejpam-976	62	10	x2	x2	PROPN
ejpam-976	62	11	,	,	PUNCT
ejpam-976	62	12	.....	.....	PUNCT
ejpam-976	62	13	,	,	PUNCT
ejpam-976	62	14	xm	xm	PROPN
ejpam-976	62	15	)	)	PUNCT
ejpam-976	62	16	and	and	CCONJ
ejpam-976	62	17	any	any	DET
ejpam-976	62	18	random	random	ADJ
ejpam-976	62	19	variable	variable	ADJ
ejpam-976	62	20	λ	λ	NOUN
ejpam-976	62	21	,	,	PUNCT
ejpam-976	62	22	which	which	PRON
ejpam-976	62	23	is	be	AUX
ejpam-976	62	24	assumed	assume	VERB
ejpam-976	62	25	to	to	PART
ejpam-976	62	26	be	be	AUX
ejpam-976	62	27	a	a	DET
ejpam-976	62	28	function	function	NOUN
ejpam-976	62	29	of	of	ADP
ejpam-976	62	30	x	x	PRON
ejpam-976	62	31	,	,	PUNCT
ejpam-976	62	32	we	we	PRON
ejpam-976	62	33	have	have	VERB
ejpam-976	62	34	,	,	PUNCT
ejpam-976	62	35	n∑	n∑	PROPN
ejpam-976	62	36	i=1	i=1	PROPN
ejpam-976	62	37	e[x	e[x	ADJ
ejpam-976	62	38	i|λ]≤cx	i|λ]≤cx	NOUN
ejpam-976	63	1	n∑	n∑	NOUN
ejpam-976	63	2	i=1	i=1	X
ejpam-976	64	1	x	x	PUNCT
ejpam-976	64	2	i	i	PRON
ejpam-976	64	3	proof	proof	VERB
ejpam-976	64	4	.	.	PUNCT
ejpam-976	65	1	from	from	ADP
ejpam-976	65	2	the	the	DET
ejpam-976	65	3	definition	definition	NOUN
ejpam-976	65	4	1	1	NUM
ejpam-976	65	5	we	we	PRON
ejpam-976	65	6	have	have	AUX
ejpam-976	65	7	,	,	PUNCT
ejpam-976	65	8	x	x	SYM
ejpam-976	65	9	≤cx	≤cx	NOUN
ejpam-976	65	10	y	y	PROPN
ejpam-976	65	11	if	if	SCONJ
ejpam-976	65	12	and	and	CCONJ
ejpam-976	65	13	only	only	ADV
ejpam-976	65	14	if	if	SCONJ
ejpam-976	65	15	e[φ(x	e[φ(x	NOUN
ejpam-976	65	16	)	)	PUNCT
ejpam-976	65	17	]	]	PUNCT
ejpam-976	65	18	≤	≤	NUM
ejpam-976	65	19	e[φ(y	e[φ(y	NUM
ejpam-976	65	20	)	)	PUNCT
ejpam-976	65	21	]	]	PUNCT
ejpam-976	65	22	.	.	PUNCT
ejpam-976	66	1	in	in	ADP
ejpam-976	66	2	accordance	accordance	NOUN
ejpam-976	66	3	with	with	ADP
ejpam-976	66	4	this	this	DET
ejpam-976	66	5	definition	definition	NOUN
ejpam-976	66	6	we	we	PRON
ejpam-976	66	7	need	need	VERB
ejpam-976	66	8	to	to	PART
ejpam-976	66	9	show	show	VERB
ejpam-976	66	10	that	that	SCONJ
ejpam-976	66	11	eλ[φ	eλ[φ	PROPN
ejpam-976	66	12	(	(	PUNCT
ejpam-976	66	13	n∑	n∑	NOUN
ejpam-976	66	14	i=1	i=1	PROPN
ejpam-976	66	15	e[x	e[x	ADJ
ejpam-976	66	16	i|λ])]≤cx	i|λ])]≤cx	NOUN
ejpam-976	66	17	e[φ	e[φ	X
ejpam-976	66	18	(	(	PUNCT
ejpam-976	66	19	n∑	n∑	NOUN
ejpam-976	66	20	i=1	i=1	PROPN
ejpam-976	66	21	x	x	PUNCT
ejpam-976	66	22	i	i	NOUN
ejpam-976	66	23	)	)	PUNCT
ejpam-976	66	24	]	]	PUNCT
ejpam-976	66	25	now	now	ADV
ejpam-976	66	26	,	,	PUNCT
ejpam-976	66	27	e[φ	e[φ	X
ejpam-976	66	28	(	(	PUNCT
ejpam-976	66	29	n∑	n∑	NOUN
ejpam-976	66	30	i=1	i=1	PROPN
ejpam-976	66	31	x	x	PUNCT
ejpam-976	66	32	i	i	NOUN
ejpam-976	66	33	)	)	PUNCT
ejpam-976	66	34	]	]	PUNCT
ejpam-976	67	1	=	=	PUNCT
ejpam-976	67	2	eλe[φ	eλe[φ	PROPN
ejpam-976	67	3	(	(	PUNCT
ejpam-976	67	4	n∑	n∑	NOUN
ejpam-976	67	5	i=1	i=1	PROPN
ejpam-976	67	6	x	x	PUNCT
ejpam-976	67	7	i)|λ]≥	i)|λ]≥	PROPN
ejpam-976	67	8	eλ[φ(e	eλ[φ(e	PROPN
ejpam-976	67	9	(	(	PUNCT
ejpam-976	67	10	n∑	n∑	NOUN
ejpam-976	67	11	i=1	i=1	PROPN
ejpam-976	67	12	x	x	SYM
ejpam-976	67	13	i|λ	i|λ	NOUN
ejpam-976	67	14	)	)	PUNCT
ejpam-976	67	15	)	)	PUNCT
ejpam-976	67	16	]	]	PUNCT
ejpam-976	68	1	=	=	SYM
ejpam-976	68	2	eλ[φ	eλ[φ	PROPN
ejpam-976	68	3	(	(	PUNCT
ejpam-976	68	4	n∑	n∑	NOUN
ejpam-976	68	5	i=1	i=1	PROPN
ejpam-976	68	6	e[x	e[x	ADJ
ejpam-976	68	7	i|λ	i|λ	NOUN
ejpam-976	68	8	]	]	PUNCT
ejpam-976	68	9	)	)	PUNCT
ejpam-976	68	10	]	]	PUNCT
ejpam-976	69	1	the	the	DET
ejpam-976	69	2	last	last	ADJ
ejpam-976	69	3	inequality	inequality	NOUN
ejpam-976	69	4	was	be	AUX
ejpam-976	69	5	obtained	obtain	VERB
ejpam-976	69	6	by	by	ADP
ejpam-976	69	7	jensen	jensen	PROPN
ejpam-976	69	8	’s	’s	PART
ejpam-976	69	9	inequality	inequality	NOUN
ejpam-976	69	10	,	,	PUNCT
ejpam-976	69	11	which	which	PRON
ejpam-976	69	12	states	state	VERB
ejpam-976	69	13	that	that	SCONJ
ejpam-976	69	14	for	for	ADP
ejpam-976	69	15	any	any	DET
ejpam-976	69	16	convex	convex	NOUN
ejpam-976	69	17	function	function	NOUN
ejpam-976	69	18	φ	φ	PROPN
ejpam-976	69	19	,	,	PUNCT
ejpam-976	69	20	φ(e(x	φ(e(x	PROPN
ejpam-976	69	21	)	)	PUNCT
ejpam-976	69	22	)	)	PUNCT
ejpam-976	69	23	≤	≤	NUM
ejpam-976	69	24	e(φ(x	e(φ(x	PROPN
ejpam-976	69	25	)	)	PUNCT
ejpam-976	69	26	)	)	PUNCT
ejpam-976	69	27	therefore	therefore	ADV
ejpam-976	69	28	,	,	PUNCT
ejpam-976	69	29	eλ[φ	eλ[φ	PROPN
ejpam-976	69	30	(	(	PUNCT
ejpam-976	69	31	n∑	n∑	NOUN
ejpam-976	69	32	i=1	i=1	PROPN
ejpam-976	69	33	e[x	e[x	NOUN
ejpam-976	69	34	i|λ])]≤	i|λ])]≤	X
ejpam-976	69	35	e[φ	e[φ	X
ejpam-976	69	36	(	(	PUNCT
ejpam-976	69	37	n∑	n∑	NOUN
ejpam-976	69	38	i=1	i=1	PROPN
ejpam-976	69	39	x	x	PUNCT
ejpam-976	69	40	i	i	NOUN
ejpam-976	69	41	)	)	PUNCT
ejpam-976	69	42	]	]	PUNCT
ejpam-976	70	1	hence	hence	ADV
ejpam-976	70	2	,	,	PUNCT
ejpam-976	70	3	n∑	n∑	PROPN
ejpam-976	70	4	i=1	i=1	PROPN
ejpam-976	70	5	e[x	e[x	ADJ
ejpam-976	70	6	i|λ]≤cx	i|λ]≤cx	NOUN
ejpam-976	70	7	n∑	n∑	NOUN
ejpam-976	70	8	i=1	i=1	PROPN
ejpam-976	71	1	x	x	PUNCT
ejpam-976	71	2	i	i	PRON
ejpam-976	71	3	which	which	PRON
ejpam-976	71	4	completes	complete	VERB
ejpam-976	71	5	the	the	DET
ejpam-976	71	6	proof	proof	NOUN
ejpam-976	71	7	.	.	PUNCT
ejpam-976	72	1	a	a	DET
ejpam-976	72	2	gupta	gupta	PROPN
ejpam-976	72	3	,	,	PUNCT
ejpam-976	72	4	m	m	PROPN
ejpam-976	72	5	aziz	aziz	PROPN
ejpam-976	72	6	/	/	SYM
ejpam-976	72	7	eur	eur	PROPN
ejpam-976	72	8	.	.	PUNCT
ejpam-976	73	1	j.	j.	PROPN
ejpam-976	73	2	pure	pure	PROPN
ejpam-976	73	3	appl	appl	PROPN
ejpam-976	73	4	.	.	PROPN
ejpam-976	73	5	math	math	PROPN
ejpam-976	73	6	,	,	PUNCT
ejpam-976	73	7	3	3	NUM
ejpam-976	73	8	(	(	PUNCT
ejpam-976	73	9	2010	2010	NUM
ejpam-976	73	10	)	)	PUNCT
ejpam-976	73	11	,	,	PUNCT
ejpam-976	73	12	779	779	NUM
ejpam-976	73	13	-	-	SYM
ejpam-976	73	14	785	785	NUM
ejpam-976	73	15	782	782	NUM
ejpam-976	73	16	2.1	2.1	NUM
ejpam-976	73	17	.	.	PUNCT
ejpam-976	74	1	lower	low	ADJ
ejpam-976	74	2	bound	bind	VERB
ejpam-976	74	3	approximations	approximation	NOUN
ejpam-976	74	4	of	of	ADP
ejpam-976	74	5	the	the	DET
ejpam-976	74	6	distribution	distribution	NOUN
ejpam-976	74	7	sum	sum	NOUN
ejpam-976	74	8	of	of	ADP
ejpam-976	74	9	random	random	ADJ
ejpam-976	74	10	variables	variable	NOUN
ejpam-976	74	11	with	with	ADP
ejpam-976	74	12	convex	convex	NOUN
ejpam-976	74	13	ordering	ordering	NOUN
ejpam-976	74	14	in	in	ADP
ejpam-976	74	15	this	this	DET
ejpam-976	74	16	section	section	NOUN
ejpam-976	74	17	we	we	PRON
ejpam-976	74	18	will	will	AUX
ejpam-976	74	19	describe	describe	VERB
ejpam-976	74	20	two	two	NUM
ejpam-976	74	21	examples	example	NOUN
ejpam-976	75	1	[	[	X
ejpam-976	75	2	follow	follow	NOUN
ejpam-976	75	3	from	from	ADP
ejpam-976	75	4	1	1	NUM
ejpam-976	75	5	]	]	PUNCT
ejpam-976	75	6	that	that	PRON
ejpam-976	75	7	show	show	VERB
ejpam-976	75	8	how	how	SCONJ
ejpam-976	75	9	distribution	distribution	NOUN
ejpam-976	75	10	function	function	NOUN
ejpam-976	75	11	of	of	ADP
ejpam-976	75	12	the	the	DET
ejpam-976	75	13	sum	sum	NOUN
ejpam-976	75	14	of	of	ADP
ejpam-976	75	15	random	random	ADJ
ejpam-976	75	16	variables	variable	NOUN
ejpam-976	75	17	can	can	AUX
ejpam-976	75	18	be	be	AUX
ejpam-976	75	19	approximated	approximate	VERB
ejpam-976	75	20	by	by	ADP
ejpam-976	75	21	convex	convex	ADJ
ejpam-976	75	22	order	order	NOUN
ejpam-976	75	23	of	of	ADP
ejpam-976	75	24	random	random	ADJ
ejpam-976	75	25	variable	variable	NOUN
ejpam-976	75	26	.	.	PUNCT
ejpam-976	76	1	example	example	NOUN
ejpam-976	76	2	1	1	NUM
ejpam-976	76	3	(	(	PUNCT
ejpam-976	76	4	approximation	approximation	NOUN
ejpam-976	76	5	of	of	ADP
ejpam-976	76	6	distribution	distribution	NOUN
ejpam-976	76	7	sum	sum	NOUN
ejpam-976	76	8	of	of	ADP
ejpam-976	76	9	two	two	NUM
ejpam-976	76	10	independent	independent	ADJ
ejpam-976	76	11	standard	standard	ADJ
ejpam-976	76	12	normal	normal	ADJ
ejpam-976	76	13	random	random	ADJ
ejpam-976	76	14	variables	variable	NOUN
ejpam-976	76	15	)	)	PUNCT
ejpam-976	76	16	.	.	PUNCT
ejpam-976	77	1	suppose	suppose	VERB
ejpam-976	77	2	x	x	PRON
ejpam-976	77	3	and	and	CCONJ
ejpam-976	77	4	y	y	PROPN
ejpam-976	77	5	be	be	AUX
ejpam-976	77	6	independent	independent	ADJ
ejpam-976	77	7	n(0,1	n(0,1	NOUN
ejpam-976	77	8	)	)	PUNCT
ejpam-976	77	9	random	random	ADJ
ejpam-976	77	10	variables	variable	NOUN
ejpam-976	77	11	.	.	PUNCT
ejpam-976	78	1	we	we	PRON
ejpam-976	78	2	want	want	VERB
ejpam-976	78	3	to	to	PART
ejpam-976	78	4	derive	derive	VERB
ejpam-976	78	5	lower	low	ADJ
ejpam-976	78	6	bounds	bound	NOUN
ejpam-976	78	7	for	for	ADP
ejpam-976	78	8	s	s	NOUN
ejpam-976	78	9	=	=	PUNCT
ejpam-976	78	10	x	x	PROPN
ejpam-976	79	1	+	+	NUM
ejpam-976	79	2	y	y	PROPN
ejpam-976	79	3	.	.	PUNCT
ejpam-976	80	1	in	in	ADP
ejpam-976	80	2	this	this	DET
ejpam-976	80	3	case	case	NOUN
ejpam-976	80	4	we	we	PRON
ejpam-976	80	5	know	know	VERB
ejpam-976	80	6	the	the	DET
ejpam-976	80	7	exact	exact	ADJ
ejpam-976	80	8	distribution	distribution	NOUN
ejpam-976	80	9	of	of	ADP
ejpam-976	80	10	s	s	PROPN
ejpam-976	80	11	,	,	PUNCT
ejpam-976	80	12	i.e	i.e	PRON
ejpam-976	80	13	s	s	VERB
ejpam-976	80	14	∼	∼	NOUN
ejpam-976	80	15	n(0,2	n(0,2	NOUN
ejpam-976	80	16	)	)	PUNCT
ejpam-976	80	17	.	.	PUNCT
ejpam-976	81	1	let	let	VERB
ejpam-976	81	2	us	we	PRON
ejpam-976	81	3	see	see	VERB
ejpam-976	81	4	how	how	SCONJ
ejpam-976	81	5	lower	low	ADJ
ejpam-976	81	6	bound	bind	VERB
ejpam-976	81	7	approximation	approximation	NOUN
ejpam-976	81	8	works	work	NOUN
ejpam-976	81	9	in	in	ADP
ejpam-976	81	10	this	this	DET
ejpam-976	81	11	case	case	NOUN
ejpam-976	81	12	.	.	PUNCT
ejpam-976	82	1	let	let	VERB
ejpam-976	82	2	z	z	NOUN
ejpam-976	82	3	=	=	PUNCT
ejpam-976	83	1	x	x	PUNCT
ejpam-976	84	1	+	+	CCONJ
ejpam-976	84	2	ay	ay	NOUN
ejpam-976	84	3	for	for	ADP
ejpam-976	84	4	some	some	DET
ejpam-976	84	5	real	real	ADJ
ejpam-976	84	6	a	a	PRON
ejpam-976	84	7	.	.	PUNCT
ejpam-976	85	1	then	then	ADV
ejpam-976	85	2	z	z	NOUN
ejpam-976	85	3	∼	∼	VERB
ejpam-976	85	4	n(0,1	n(0,1	PROPN
ejpam-976	85	5	+	+	NOUN
ejpam-976	85	6	a2	a2	NOUN
ejpam-976	85	7	)	)	PUNCT
ejpam-976	85	8	.	.	PUNCT
ejpam-976	86	1	the	the	DET
ejpam-976	86	2	conditional	conditional	ADJ
ejpam-976	86	3	distribution	distribution	NOUN
ejpam-976	86	4	of	of	ADP
ejpam-976	86	5	s|z	s|z	NOUN
ejpam-976	86	6	is	be	AUX
ejpam-976	86	7	n[µs	n[µs	PROPN
ejpam-976	86	8	+	+	PROPN
ejpam-976	86	9	ρs	ρs	ADV
ejpam-976	86	10	,	,	PUNCT
ejpam-976	86	11	zσs	zσs	X
ejpam-976	86	12	σz	σz	PROPN
ejpam-976	86	13	(	(	PUNCT
ejpam-976	86	14	z	z	NOUN
ejpam-976	86	15	−µz	−µz	PROPN
ejpam-976	86	16	)	)	PUNCT
ejpam-976	86	17	,	,	PUNCT
ejpam-976	86	18	σ	σ	PROPN
ejpam-976	86	19	2	2	NUM
ejpam-976	86	20	s(1−ρ2	s(1−ρ2	PROPN
ejpam-976	86	21	s	s	PROPN
ejpam-976	86	22	,	,	PUNCT
ejpam-976	86	23	z	z	NOUN
ejpam-976	86	24	)	)	PUNCT
ejpam-976	86	25	]	]	PUNCT
ejpam-976	86	26	here	here	ADV
ejpam-976	86	27	cov(x+y	cov(x+y	PROPN
ejpam-976	86	28	,	,	PUNCT
ejpam-976	86	29	x+ay	x+ay	PROPN
ejpam-976	86	30	)	)	PUNCT
ejpam-976	87	1	=	=	SYM
ejpam-976	87	2	1.1cov(x	1.1cov(x	NUM
ejpam-976	87	3	,	,	PUNCT
ejpam-976	87	4	x	x	PROPN
ejpam-976	87	5	)	)	PUNCT
ejpam-976	88	1	+1.acov(y	+1.acov(y	PROPN
ejpam-976	88	2	,	,	PUNCT
ejpam-976	88	3	y	y	PROPN
ejpam-976	88	4	)	)	PUNCT
ejpam-976	89	1	=	=	PUNCT
ejpam-976	89	2	1+a	1+a	NUM
ejpam-976	89	3	and	and	CCONJ
ejpam-976	89	4	ρs	ρs	ADV
ejpam-976	89	5	,	,	PUNCT
ejpam-976	89	6	z	z	NOUN
ejpam-976	89	7	=	=	PUNCT
ejpam-976	90	1	1+ap	1+ap	NUM
ejpam-976	90	2	2	2	NUM
ejpam-976	90	3	p	p	NOUN
ejpam-976	90	4	1+a2	1+a2	NUM
ejpam-976	90	5	.	.	PUNCT
ejpam-976	91	1	therefore	therefore	ADV
ejpam-976	91	2	,	,	PUNCT
ejpam-976	91	3	s|z	s|z	ADV
ejpam-976	91	4	∼	∼	NOUN
ejpam-976	91	5	n[z	n[z	ADP
ejpam-976	91	6	1	1	NUM
ejpam-976	91	7	+	+	CCONJ
ejpam-976	91	8	a	a	DET
ejpam-976	91	9	1	1	NUM
ejpam-976	91	10	+	+	NUM
ejpam-976	91	11	a2	a2	NOUN
ejpam-976	91	12	,	,	PUNCT
ejpam-976	91	13	(	(	PUNCT
ejpam-976	91	14	1−	1−	NUM
ejpam-976	91	15	a)2	a)2	NOUN
ejpam-976	91	16	1	1	NUM
ejpam-976	91	17	+	+	NUM
ejpam-976	91	18	a2	a2	PROPN
ejpam-976	91	19	]	]	PUNCT
ejpam-976	91	20	hence	hence	ADV
ejpam-976	91	21	e[s|z	e[s|z	NOUN
ejpam-976	91	22	]	]	X
ejpam-976	91	23	=	=	SYM
ejpam-976	91	24	1	1	NUM
ejpam-976	91	25	+	+	CCONJ
ejpam-976	91	26	a	a	DET
ejpam-976	91	27	1	1	NUM
ejpam-976	91	28	+	+	NUM
ejpam-976	91	29	a2	a2	PROPN
ejpam-976	91	30	z	z	PROPN
ejpam-976	91	31	∼	∼	NOUN
ejpam-976	91	32	n[0	n[0	PROPN
ejpam-976	91	33	,	,	PUNCT
ejpam-976	91	34	(	(	PUNCT
ejpam-976	91	35	1	1	NUM
ejpam-976	91	36	+	+	NUM
ejpam-976	91	37	a)2	a)2	NOUN
ejpam-976	91	38	1	1	NUM
ejpam-976	91	39	+	+	NUM
ejpam-976	91	40	a2	a2	PROPN
ejpam-976	91	41	]	]	PUNCT
ejpam-976	91	42	therefore	therefore	ADV
ejpam-976	91	43	,	,	PUNCT
ejpam-976	91	44	for	for	ADP
ejpam-976	91	45	some	some	DET
ejpam-976	91	46	choices	choice	NOUN
ejpam-976	91	47	of	of	ADP
ejpam-976	91	48	a	a	PRON
ejpam-976	91	49	,	,	PUNCT
ejpam-976	91	50	we	we	PRON
ejpam-976	91	51	get	get	VERB
ejpam-976	91	52	the	the	DET
ejpam-976	91	53	following	follow	VERB
ejpam-976	91	54	distribution	distribution	NOUN
ejpam-976	91	55	for	for	ADP
ejpam-976	91	56	the	the	DET
ejpam-976	91	57	lower	low	ADJ
ejpam-976	91	58	bound	bind	VERB
ejpam-976	91	59	for	for	ADP
ejpam-976	91	60	s	s	NOUN
ejpam-976	91	61	:	:	PUNCT
ejpam-976	91	62	a	a	DET
ejpam-976	91	63	=	=	SYM
ejpam-976	91	64	0	0	NUM
ejpam-976	91	65	gives	give	VERB
ejpam-976	91	66	n(0,1)≤cx	n(0,1)≤cx	NOUN
ejpam-976	91	67	s	s	PART
ejpam-976	91	68	=	=	PUNCT
ejpam-976	91	69	x	x	SYM
ejpam-976	91	70	+	+	PUNCT
ejpam-976	91	71	y	y	PROPN
ejpam-976	91	72	∼	∼	NOUN
ejpam-976	91	73	n(0,2	n(0,2	NOUN
ejpam-976	91	74	)	)	PUNCT
ejpam-976	91	75	a	a	DET
ejpam-976	91	76	=	=	SYM
ejpam-976	91	77	1	1	NUM
ejpam-976	91	78	gives	give	VERB
ejpam-976	91	79	n(0,2)≤cx	n(0,2)≤cx	NOUN
ejpam-976	91	80	s	s	PART
ejpam-976	91	81	=	=	PUNCT
ejpam-976	91	82	x	x	SYM
ejpam-976	92	1	+	+	PUNCT
ejpam-976	92	2	y	y	PROPN
ejpam-976	92	3	∼	∼	NOUN
ejpam-976	92	4	n(0,2	n(0,2	NOUN
ejpam-976	92	5	)	)	PUNCT
ejpam-976	92	6	a	a	DET
ejpam-976	92	7	=	=	NOUN
ejpam-976	92	8	−1	−1	NOUN
ejpam-976	92	9	gives	give	VERB
ejpam-976	92	10	n(0,0)≤cx	n(0,0)≤cx	PROPN
ejpam-976	92	11	s	s	PART
ejpam-976	92	12	=	=	PUNCT
ejpam-976	92	13	x	x	SYM
ejpam-976	93	1	+	+	PUNCT
ejpam-976	93	2	y	y	PROPN
ejpam-976	93	3	∼	∼	NOUN
ejpam-976	93	4	n(0,2	n(0,2	NOUN
ejpam-976	93	5	)	)	PUNCT
ejpam-976	93	6	thus	thus	ADV
ejpam-976	93	7	in	in	ADP
ejpam-976	93	8	this	this	DET
ejpam-976	93	9	case	case	NOUN
ejpam-976	93	10	best	well	ADV
ejpam-976	93	11	lower	lower	ADV
ejpam-976	93	12	bound	bind	VERB
ejpam-976	93	13	is	be	AUX
ejpam-976	93	14	obtained	obtain	VERB
ejpam-976	93	15	for	for	ADP
ejpam-976	93	16	a	a	DET
ejpam-976	93	17	=	=	SYM
ejpam-976	93	18	1	1	NUM
ejpam-976	93	19	which	which	PRON
ejpam-976	93	20	is	be	AUX
ejpam-976	93	21	the	the	DET
ejpam-976	93	22	exact	exact	ADJ
ejpam-976	93	23	distribution	distribution	NOUN
ejpam-976	93	24	.	.	PUNCT
ejpam-976	94	1	the	the	DET
ejpam-976	94	2	variance	variance	NOUN
ejpam-976	94	3	of	of	ADP
ejpam-976	94	4	the	the	DET
ejpam-976	94	5	lower	low	ADJ
ejpam-976	94	6	bound	bind	VERB
ejpam-976	94	7	can	can	AUX
ejpam-976	94	8	be	be	AUX
ejpam-976	94	9	seen	see	VERB
ejpam-976	94	10	to	to	PART
ejpam-976	94	11	have	have	VERB
ejpam-976	94	12	a	a	DET
ejpam-976	94	13	maximum	maximum	NOUN
ejpam-976	94	14	at	at	ADP
ejpam-976	94	15	a	a	DET
ejpam-976	94	16	=	=	SYM
ejpam-976	94	17	1	1	NUM
ejpam-976	94	18	and	and	CCONJ
ejpam-976	94	19	a	a	DET
ejpam-976	94	20	minimum	minimum	NOUN
ejpam-976	94	21	at	at	ADP
ejpam-976	94	22	a	a	DET
ejpam-976	94	23	=	=	NOUN
ejpam-976	94	24	−1	−1	NOUN
ejpam-976	94	25	.	.	PUNCT
ejpam-976	95	1	example	example	NOUN
ejpam-976	95	2	2	2	NUM
ejpam-976	95	3	(	(	PUNCT
ejpam-976	95	4	approximation	approximation	NOUN
ejpam-976	95	5	of	of	ADP
ejpam-976	95	6	distribution	distribution	NOUN
ejpam-976	95	7	sum	sum	NOUN
ejpam-976	95	8	of	of	ADP
ejpam-976	95	9	two	two	NUM
ejpam-976	95	10	lognormal	lognormal	ADJ
ejpam-976	95	11	random	random	ADJ
ejpam-976	95	12	variables	variable	NOUN
ejpam-976	95	13	)	)	PUNCT
ejpam-976	95	14	.	.	PUNCT
ejpam-976	96	1	suppose	suppose	VERB
ejpam-976	96	2	y1	y1	INTJ
ejpam-976	96	3	and	and	CCONJ
ejpam-976	96	4	y2	y2	NOUN
ejpam-976	96	5	be	be	AUX
ejpam-976	96	6	independent	independent	ADJ
ejpam-976	96	7	n(0,1	n(0,1	NOUN
ejpam-976	96	8	)	)	PUNCT
ejpam-976	96	9	.	.	PUNCT
ejpam-976	97	1	define	define	VERB
ejpam-976	97	2	x1	x1	NOUN
ejpam-976	97	3	=	=	SYM
ejpam-976	98	1	ey1	ey1	ADJ
ejpam-976	98	2	⇒	⇒	VERB
ejpam-976	98	3	x1	x1	ADJ
ejpam-976	98	4	∼	∼	NOUN
ejpam-976	98	5	lognormal(0,1	lognormal(0,1	NOUN
ejpam-976	98	6	)	)	PUNCT
ejpam-976	98	7	and	and	CCONJ
ejpam-976	98	8	x2	x2	NOUN
ejpam-976	98	9	=	=	PUNCT
ejpam-976	98	10	ey1+y2	ey1+y2	PROPN
ejpam-976	98	11	⇒	⇒	VERB
ejpam-976	98	12	x2	x2	ADJ
ejpam-976	98	13	∼	∼	NOUN
ejpam-976	98	14	lognormal(0,2	lognormal(0,2	NOUN
ejpam-976	98	15	)	)	PUNCT
ejpam-976	98	16	.	.	PUNCT
ejpam-976	99	1	we	we	PRON
ejpam-976	99	2	want	want	VERB
ejpam-976	99	3	to	to	PART
ejpam-976	99	4	find	find	VERB
ejpam-976	99	5	the	the	DET
ejpam-976	99	6	lower	lower	ADV
ejpam-976	99	7	bound	bind	VERB
ejpam-976	99	8	for	for	ADP
ejpam-976	99	9	the	the	DET
ejpam-976	99	10	distribution	distribution	NOUN
ejpam-976	99	11	of	of	ADP
ejpam-976	99	12	s	s	NOUN
ejpam-976	99	13	=	=	PUNCT
ejpam-976	99	14	x1	x1	PROPN
ejpam-976	100	1	+	+	CCONJ
ejpam-976	100	2	x2	x2	PROPN
ejpam-976	100	3	.	.	PUNCT
ejpam-976	101	1	let	let	VERB
ejpam-976	101	2	z	z	NOUN
ejpam-976	101	3	=	=	PUNCT
ejpam-976	101	4	y1	y1	PROPN
ejpam-976	101	5	+	+	CCONJ
ejpam-976	101	6	y2	y2	NOUN
ejpam-976	101	7	.	.	PUNCT
ejpam-976	102	1	as	as	SCONJ
ejpam-976	102	2	shown	show	VERB
ejpam-976	102	3	in	in	ADP
ejpam-976	102	4	example	example	NOUN
ejpam-976	102	5	1	1	NUM
ejpam-976	102	6	,	,	PUNCT
ejpam-976	102	7	the	the	DET
ejpam-976	102	8	conditional	conditional	ADJ
ejpam-976	102	9	distribution	distribution	NOUN
ejpam-976	102	10	of	of	ADP
ejpam-976	102	11	y1	y1	NOUN
ejpam-976	102	12	given	give	VERB
ejpam-976	102	13	z	z	PROPN
ejpam-976	102	14	is	be	AUX
ejpam-976	102	15	,	,	PUNCT
ejpam-976	102	16	y1|(y1	y1|(y1	PROPN
ejpam-976	102	17	+	+	CCONJ
ejpam-976	102	18	y2	y2	NOUN
ejpam-976	102	19	)	)	PUNCT
ejpam-976	103	1	=	=	SYM
ejpam-976	103	2	z	z	NOUN
ejpam-976	103	3	∼	∼	NOUN
ejpam-976	103	4	n	n	CCONJ
ejpam-976	103	5	(	(	PUNCT
ejpam-976	103	6	1	1	NUM
ejpam-976	103	7	2	2	NUM
ejpam-976	103	8	z	z	NOUN
ejpam-976	103	9	,	,	PUNCT
ejpam-976	103	10	1	1	NUM
ejpam-976	103	11	2	2	NUM
ejpam-976	103	12	)	)	PUNCT
ejpam-976	103	13	therefore	therefore	ADV
ejpam-976	103	14	,	,	PUNCT
ejpam-976	103	15	e[x1	e[x1	ADP
ejpam-976	103	16	=	=	PUNCT
ejpam-976	103	17	ey1	ey1	ADJ
ejpam-976	103	18	|(y1	|(y1	ADJ
ejpam-976	103	19	+	+	CCONJ
ejpam-976	103	20	y2	y2	NOUN
ejpam-976	103	21	)	)	PUNCT
ejpam-976	104	1	=	=	PUNCT
ejpam-976	105	1	z	z	X
ejpam-976	105	2	]	]	X
ejpam-976	105	3	=	=	PUNCT
ejpam-976	105	4	my	my	PRON
ejpam-976	105	5	(	(	PUNCT
ejpam-976	105	6	1	1	NUM
ejpam-976	105	7	,	,	PUNCT
ejpam-976	105	8	1	1	NUM
ejpam-976	105	9	2	2	NUM
ejpam-976	105	10	z	z	NOUN
ejpam-976	105	11	,	,	PUNCT
ejpam-976	105	12	1	1	NUM
ejpam-976	105	13	2	2	NUM
ejpam-976	105	14	)	)	PUNCT
ejpam-976	105	15	,	,	PUNCT
ejpam-976	105	16	where	where	SCONJ
ejpam-976	105	17	y	y	PROPN
ejpam-976	105	18	∼	∼	VERB
ejpam-976	105	19	n(µ,σ2	n(µ,σ2	NOUN
ejpam-976	105	20	)	)	PUNCT
ejpam-976	105	21	=	=	SYM
ejpam-976	106	1	ex	ex	PRON
ejpam-976	106	2	p	p	X
ejpam-976	106	3	(	(	PUNCT
ejpam-976	106	4	1	1	NUM
ejpam-976	106	5	2	2	NUM
ejpam-976	106	6	z	z	NOUN
ejpam-976	106	7	+	+	NOUN
ejpam-976	106	8	1	1	NUM
ejpam-976	106	9	4	4	NUM
ejpam-976	106	10	)	)	PUNCT
ejpam-976	106	11	a	a	DET
ejpam-976	106	12	gupta	gupta	PROPN
ejpam-976	106	13	,	,	PUNCT
ejpam-976	106	14	m	m	PROPN
ejpam-976	106	15	aziz	aziz	PROPN
ejpam-976	106	16	/	/	SYM
ejpam-976	106	17	eur	eur	PROPN
ejpam-976	106	18	.	.	PUNCT
ejpam-976	107	1	j.	j.	PROPN
ejpam-976	107	2	pure	pure	PROPN
ejpam-976	107	3	appl	appl	PROPN
ejpam-976	107	4	.	.	PROPN
ejpam-976	107	5	math	math	PROPN
ejpam-976	107	6	,	,	PUNCT
ejpam-976	107	7	3	3	NUM
ejpam-976	107	8	(	(	PUNCT
ejpam-976	107	9	2010	2010	NUM
ejpam-976	107	10	)	)	PUNCT
ejpam-976	107	11	,	,	PUNCT
ejpam-976	107	12	779	779	NUM
ejpam-976	107	13	-	-	SYM
ejpam-976	107	14	785	785	NUM
ejpam-976	107	15	783	783	NUM
ejpam-976	107	16	we	we	PRON
ejpam-976	107	17	also	also	ADV
ejpam-976	107	18	observe	observe	VERB
ejpam-976	107	19	that	that	SCONJ
ejpam-976	107	20	e[x2|(y1	e[x2|(y1	PROPN
ejpam-976	107	21	+	+	NOUN
ejpam-976	107	22	y2	y2	NOUN
ejpam-976	107	23	)	)	PUNCT
ejpam-976	108	1	=	=	PUNCT
ejpam-976	109	1	z	z	X
ejpam-976	109	2	]	]	X
ejpam-976	109	3	=	=	SYM
ejpam-976	109	4	ez	ez	PROPN
ejpam-976	109	5	.therefore	.therefore	PUNCT
ejpam-976	109	6	the	the	DET
ejpam-976	109	7	lower	lower	ADV
ejpam-976	109	8	bound	bind	VERB
ejpam-976	109	9	for	for	ADP
ejpam-976	109	10	approximating	approximate	VERB
ejpam-976	109	11	the	the	DET
ejpam-976	109	12	distribution	distribution	NOUN
ejpam-976	109	13	of	of	ADP
ejpam-976	109	14	s	s	NOUN
ejpam-976	109	15	=	=	SYM
ejpam-976	109	16	x1	x1	PROPN
ejpam-976	110	1	+	+	CCONJ
ejpam-976	110	2	x2	x2	NOUN
ejpam-976	110	3	is	be	AUX
ejpam-976	110	4	sl	sl	INTJ
ejpam-976	110	5	=	=	SYM
ejpam-976	110	6	e[(x1	e[(x1	PROPN
ejpam-976	111	1	+	+	NUM
ejpam-976	111	2	x2)|z	x2)|z	X
ejpam-976	111	3	]	]	X
ejpam-976	111	4	=	=	PUNCT
ejpam-976	112	1	ex	ex	X
ejpam-976	112	2	p(1	p(1	NOUN
ejpam-976	112	3	2	2	NUM
ejpam-976	112	4	z	z	NOUN
ejpam-976	112	5	+	+	NOUN
ejpam-976	112	6	1	1	NUM
ejpam-976	112	7	4	4	NUM
ejpam-976	112	8	)	)	PUNCT
ejpam-976	112	9	.	.	PUNCT
ejpam-976	113	1	it	it	PRON
ejpam-976	113	2	can	can	AUX
ejpam-976	113	3	be	be	AUX
ejpam-976	113	4	easily	easily	ADV
ejpam-976	113	5	verified	verify	VERB
ejpam-976	113	6	that	that	SCONJ
ejpam-976	113	7	e(sl	e(sl	X
ejpam-976	113	8	)	)	PUNCT
ejpam-976	114	1	=	=	PUNCT
ejpam-976	114	2	e[ex	e[ex	PROPN
ejpam-976	114	3	p(1	p(1	PROPN
ejpam-976	114	4	2	2	NUM
ejpam-976	114	5	z	z	NOUN
ejpam-976	114	6	+	+	NOUN
ejpam-976	114	7	1	1	NUM
ejpam-976	114	8	4	4	NUM
ejpam-976	114	9	)	)	PUNCT
ejpam-976	114	10	]	]	PUNCT
ejpam-976	115	1	=	=	PUNCT
ejpam-976	115	2	e	e	X
ejpam-976	115	3	1	1	NUM
ejpam-976	115	4	2	2	NUM
ejpam-976	115	5	+	+	CCONJ
ejpam-976	115	6	e	e	NOUN
ejpam-976	115	7	and	and	CCONJ
ejpam-976	115	8	e((sl	e((sl	NUM
ejpam-976	115	9	)	)	PUNCT
ejpam-976	115	10	2	2	NUM
ejpam-976	115	11	)	)	PUNCT
ejpam-976	115	12	=	=	PUNCT
ejpam-976	115	13	e	e	NOUN
ejpam-976	115	14	3	3	NUM
ejpam-976	115	15	2	2	NUM
ejpam-976	115	16	+	+	NUM
ejpam-976	115	17	2e	2e	NUM
ejpam-976	115	18	5	5	NUM
ejpam-976	115	19	2	2	NUM
ejpam-976	115	20	+	+	CCONJ
ejpam-976	115	21	e4	e4	PROPN
ejpam-976	115	22	.	.	PUNCT
ejpam-976	116	1	thus	thus	ADV
ejpam-976	116	2	the	the	DET
ejpam-976	116	3	variance	variance	NOUN
ejpam-976	116	4	of	of	ADP
ejpam-976	116	5	the	the	DET
ejpam-976	116	6	lower	lower	ADV
ejpam-976	116	7	bound	bind	VERB
ejpam-976	116	8	is	be	AUX
ejpam-976	116	9	64.374	64.374	NUM
ejpam-976	116	10	and	and	CCONJ
ejpam-976	116	11	is	be	AUX
ejpam-976	116	12	close	close	ADJ
ejpam-976	116	13	to	to	ADP
ejpam-976	116	14	the	the	DET
ejpam-976	116	15	variance	variance	NOUN
ejpam-976	116	16	of	of	ADP
ejpam-976	116	17	s	s	NOUN
ejpam-976	116	18	=	=	SYM
ejpam-976	116	19	67.281	67.281	NUM
ejpam-976	116	20	.	.	PUNCT
ejpam-976	117	1	the	the	DET
ejpam-976	117	2	idea	idea	NOUN
ejpam-976	117	3	is	be	AUX
ejpam-976	117	4	to	to	PART
ejpam-976	117	5	obtain	obtain	VERB
ejpam-976	117	6	lower	low	ADJ
ejpam-976	117	7	convex	convex	NOUN
ejpam-976	117	8	bound	bind	VERB
ejpam-976	117	9	in	in	ADP
ejpam-976	117	10	such	such	DET
ejpam-976	117	11	a	a	DET
ejpam-976	117	12	way	way	NOUN
ejpam-976	117	13	that	that	PRON
ejpam-976	117	14	the	the	DET
ejpam-976	117	15	variance	variance	NOUN
ejpam-976	117	16	of	of	ADP
ejpam-976	117	17	the	the	DET
ejpam-976	117	18	lower	low	ADJ
ejpam-976	117	19	bound	bind	VERB
ejpam-976	117	20	gets	get	VERB
ejpam-976	117	21	as	as	ADV
ejpam-976	117	22	close	close	ADJ
ejpam-976	117	23	as	as	ADP
ejpam-976	117	24	possible	possible	ADJ
ejpam-976	117	25	to	to	ADP
ejpam-976	117	26	the	the	DET
ejpam-976	117	27	variance	variance	NOUN
ejpam-976	117	28	of	of	ADP
ejpam-976	117	29	the	the	DET
ejpam-976	117	30	sum	sum	NOUN
ejpam-976	117	31	.	.	PUNCT
ejpam-976	118	1	with	with	ADP
ejpam-976	118	2	this	this	DET
ejpam-976	118	3	view	view	NOUN
ejpam-976	118	4	in	in	ADP
ejpam-976	118	5	mind	mind	NOUN
ejpam-976	118	6	considering	consider	VERB
ejpam-976	118	7	more	more	ADJ
ejpam-976	118	8	general	general	ADJ
ejpam-976	118	9	form	form	NOUN
ejpam-976	118	10	of	of	ADP
ejpam-976	118	11	the	the	DET
ejpam-976	118	12	conditioning	conditioning	NOUN
ejpam-976	118	13	variable	variable	NOUN
ejpam-976	118	14	as	as	ADP
ejpam-976	118	15	z	z	NOUN
ejpam-976	118	16	=	=	SYM
ejpam-976	118	17	y1	y1	PROPN
ejpam-976	118	18	+	+	CCONJ
ejpam-976	118	19	ay2	ay2	PROPN
ejpam-976	118	20	,	,	PUNCT
ejpam-976	118	21	it	it	PRON
ejpam-976	118	22	could	could	AUX
ejpam-976	118	23	be	be	AUX
ejpam-976	118	24	shown	show	VERB
ejpam-976	118	25	that	that	SCONJ
ejpam-976	118	26	optimal	optimal	ADJ
ejpam-976	118	27	lower	low	ADJ
ejpam-976	118	28	bound	bind	VERB
ejpam-976	118	29	is	be	AUX
ejpam-976	118	30	reached	reach	VERB
ejpam-976	118	31	for	for	ADP
ejpam-976	118	32	a	a	DET
ejpam-976	118	33	=	=	SYM
ejpam-976	118	34	1.27	1.27	NUM
ejpam-976	118	35	and	and	CCONJ
ejpam-976	118	36	the	the	DET
ejpam-976	118	37	variance	variance	NOUN
ejpam-976	118	38	of	of	ADP
ejpam-976	118	39	sl	sl	NOUN
ejpam-976	118	40	in	in	ADP
ejpam-976	118	41	this	this	DET
ejpam-976	118	42	case	case	NOUN
ejpam-976	118	43	is	be	AUX
ejpam-976	118	44	66.082	66.082	NUM
ejpam-976	118	45	.	.	PUNCT
ejpam-976	119	1	thus	thus	ADV
ejpam-976	119	2	choosing	choose	VERB
ejpam-976	119	3	the	the	DET
ejpam-976	119	4	conditioning	conditioning	NOUN
ejpam-976	119	5	variable	variable	NOUN
ejpam-976	119	6	is	be	AUX
ejpam-976	119	7	crucial	crucial	ADJ
ejpam-976	119	8	in	in	ADP
ejpam-976	119	9	determining	determine	VERB
ejpam-976	119	10	the	the	DET
ejpam-976	119	11	lower	low	ADJ
ejpam-976	119	12	convex	convex	NOUN
ejpam-976	119	13	order	order	NOUN
ejpam-976	119	14	bound	bind	VERB
ejpam-976	119	15	.	.	PUNCT
ejpam-976	120	1	3	3	X
ejpam-976	120	2	.	.	X
ejpam-976	120	3	application	application	NOUN
ejpam-976	120	4	of	of	ADP
ejpam-976	120	5	convex	convex	ADJ
ejpam-976	120	6	order	order	NOUN
ejpam-976	120	7	of	of	ADP
ejpam-976	120	8	random	random	ADJ
ejpam-976	120	9	variables	variable	NOUN
ejpam-976	120	10	in	in	ADP
ejpam-976	120	11	finance	finance	NOUN
ejpam-976	120	12	and	and	CCONJ
ejpam-976	120	13	economics	economic	NOUN
ejpam-976	120	14	let	let	VERB
ejpam-976	120	15	α0,α1,α2	α0,α1,α2	NOUN
ejpam-976	120	16	,	,	PUNCT
ejpam-976	120	17	.	.	PUNCT
ejpam-976	120	18	.	.	PUNCT
ejpam-976	120	19	.	.	PUNCT
ejpam-976	121	1	,	,	PUNCT
ejpam-976	121	2	αn−1	αn−1	ADV
ejpam-976	121	3	be	be	AUX
ejpam-976	121	4	non	non	ADJ
ejpam-976	121	5	-	-	ADJ
ejpam-976	121	6	negative	negative	ADJ
ejpam-976	121	7	real	real	ADJ
ejpam-976	121	8	numbers	number	NOUN
ejpam-976	121	9	.	.	PUNCT
ejpam-976	122	1	let	let	VERB
ejpam-976	122	2	y	y	PROPN
ejpam-976	122	3	=	=	SYM
ejpam-976	122	4	(	(	PUNCT
ejpam-976	122	5	y1	y1	PROPN
ejpam-976	122	6	,	,	PUNCT
ejpam-976	122	7	y2	y2	PROPN
ejpam-976	122	8	,	,	PUNCT
ejpam-976	122	9	.	.	PUNCT
ejpam-976	122	10	.	.	PUNCT
ejpam-976	123	1	.	.	PUNCT
ejpam-976	124	1	,	,	PUNCT
ejpam-976	124	2	yn	yn	PROPN
ejpam-976	124	3	)	)	PUNCT
ejpam-976	124	4	t	t	AUX
ejpam-976	124	5	be	be	AUX
ejpam-976	124	6	a	a	DET
ejpam-976	124	7	multivariate	multivariate	NOUN
ejpam-976	124	8	skew	skew	ADJ
ejpam-976	124	9	normal	normal	ADJ
ejpam-976	124	10	random	random	ADJ
ejpam-976	124	11	vector	vector	NOUN
ejpam-976	124	12	with	with	ADP
ejpam-976	124	13	specified	specified	ADJ
ejpam-976	124	14	mean	mean	ADJ
ejpam-976	124	15	vector	vector	NOUN
ejpam-976	124	16	and	and	CCONJ
ejpam-976	124	17	variance	variance	NOUN
ejpam-976	124	18	-	-	PUNCT
ejpam-976	124	19	covariance	covariance	NOUN
ejpam-976	124	20	matrix	matrix	NOUN
ejpam-976	124	21	and	and	CCONJ
ejpam-976	124	22	satisfying	satisfy	VERB
ejpam-976	124	23	additive	additive	ADJ
ejpam-976	124	24	properties	property	NOUN
ejpam-976	124	25	.	.	PUNCT
ejpam-976	125	1	define	define	NOUN
ejpam-976	125	2	,	,	PUNCT
ejpam-976	125	3	zi	zi	NOUN
ejpam-976	125	4	=	=	PUNCT
ejpam-976	126	1	∑n	∑n	PROPN
ejpam-976	127	1	k	k	NOUN
ejpam-976	127	2	=	=	PROPN
ejpam-976	127	3	i+1	i+1	NOUN
ejpam-976	127	4	yk	yk	PROPN
ejpam-976	127	5	,	,	PUNCT
ejpam-976	127	6	i	i	NOUN
ejpam-976	127	7	=	=	NOUN
ejpam-976	127	8	0,1	0,1	NUM
ejpam-976	127	9	,	,	PUNCT
ejpam-976	127	10	.	.	PUNCT
ejpam-976	127	11	.	.	PUNCT
ejpam-976	128	1	.	.	PUNCT
ejpam-976	129	1	,	,	PUNCT
ejpam-976	129	2	n−	n−	NOUN
ejpam-976	129	3	1	1	NUM
ejpam-976	129	4	,	,	PUNCT
ejpam-976	129	5	that	that	ADV
ejpam-976	129	6	is	is	ADV
ejpam-976	129	7	,	,	PUNCT
ejpam-976	129	8	zi	zi	PROPN
ejpam-976	129	9	’s	’	VERB
ejpam-976	129	10	are	be	AUX
ejpam-976	129	11	linear	linear	ADJ
ejpam-976	129	12	combinations	combination	NOUN
ejpam-976	129	13	of	of	ADP
ejpam-976	129	14	the	the	DET
ejpam-976	129	15	components	component	NOUN
ejpam-976	129	16	(	(	PUNCT
ejpam-976	129	17	y1	y1	INTJ
ejpam-976	129	18	,	,	PUNCT
ejpam-976	129	19	y2	y2	PROPN
ejpam-976	129	20	,	,	PUNCT
ejpam-976	129	21	.	.	PUNCT
ejpam-976	129	22	.	.	PUNCT
ejpam-976	130	1	.	.	PUNCT
ejpam-976	131	1	,	,	PUNCT
ejpam-976	131	2	yn	yn	PROPN
ejpam-976	131	3	)	)	PUNCT
ejpam-976	131	4	.	.	PUNCT
ejpam-976	132	1	with	with	ADP
ejpam-976	132	2	the	the	DET
ejpam-976	132	3	components	component	NOUN
ejpam-976	132	4	so	so	ADV
ejpam-976	132	5	defined	define	VERB
ejpam-976	132	6	,	,	PUNCT
ejpam-976	132	7	consider	consider	VERB
ejpam-976	132	8	the	the	DET
ejpam-976	132	9	sum	sum	NOUN
ejpam-976	132	10	s	s	PART
ejpam-976	132	11	=	=	PUNCT
ejpam-976	132	12	n−1∑	n−1∑	PROPN
ejpam-976	132	13	i=0	i=0	PROPN
ejpam-976	132	14	αie	αie	X
ejpam-976	132	15	zi	zi	PROPN
ejpam-976	132	16	=	=	PUNCT
ejpam-976	133	1	n−1∑	n−1∑	PROPN
ejpam-976	133	2	i=0	i=0	PROPN
ejpam-976	133	3	αie	αie	NUM
ejpam-976	133	4	yi+1+,	yi+1+,	VERB
ejpam-976	133	5	...	...	PUNCT
ejpam-976	133	6	,+yn	,+yn	PUNCT
ejpam-976	133	7	(	(	PUNCT
ejpam-976	133	8	1	1	NUM
ejpam-976	133	9	)	)	PUNCT
ejpam-976	133	10	from	from	ADP
ejpam-976	133	11	economic	economic	ADJ
ejpam-976	133	12	or	or	CCONJ
ejpam-976	133	13	actuarial	actuarial	ADJ
ejpam-976	133	14	point	point	NOUN
ejpam-976	133	15	of	of	ADP
ejpam-976	133	16	view	view	NOUN
ejpam-976	133	17	,	,	PUNCT
ejpam-976	133	18	the	the	DET
ejpam-976	133	19	sum	sum	NOUN
ejpam-976	133	20	s	s	VERB
ejpam-976	133	21	could	could	AUX
ejpam-976	133	22	be	be	AUX
ejpam-976	133	23	interpreted	interpret	VERB
ejpam-976	133	24	as	as	ADP
ejpam-976	133	25	the	the	DET
ejpam-976	133	26	final	final	ADJ
ejpam-976	133	27	wealth	wealth	NOUN
ejpam-976	133	28	or	or	CCONJ
ejpam-976	133	29	terminal	terminal	ADJ
ejpam-976	133	30	wealth	wealth	NOUN
ejpam-976	133	31	or	or	CCONJ
ejpam-976	133	32	the	the	DET
ejpam-976	133	33	accumulated	accumulate	VERB
ejpam-976	133	34	value	value	NOUN
ejpam-976	133	35	of	of	ADP
ejpam-976	133	36	a	a	DET
ejpam-976	133	37	series	series	NOUN
ejpam-976	133	38	of	of	ADP
ejpam-976	133	39	deterministic	deterministic	ADJ
ejpam-976	133	40	saving	saving	NOUN
ejpam-976	133	41	amounts	amount	NOUN
ejpam-976	133	42	or	or	CCONJ
ejpam-976	133	43	alternatively	alternatively	ADV
ejpam-976	133	44	the	the	DET
ejpam-976	133	45	accumulated	accumulate	VERB
ejpam-976	133	46	value	value	NOUN
ejpam-976	133	47	of	of	ADP
ejpam-976	133	48	a	a	DET
ejpam-976	133	49	series	series	NOUN
ejpam-976	133	50	of	of	ADP
ejpam-976	133	51	payments	payment	NOUN
ejpam-976	133	52	.	.	PUNCT
ejpam-976	134	1	in	in	ADP
ejpam-976	134	2	this	this	DET
ejpam-976	134	3	situation	situation	NOUN
ejpam-976	134	4	,	,	PUNCT
ejpam-976	134	5	αi	αi	VERB
ejpam-976	134	6	(	(	PUNCT
ejpam-976	134	7	i	i	NOUN
ejpam-976	134	8	=	=	NOUN
ejpam-976	134	9	0	0	NUM
ejpam-976	134	10	,	,	PUNCT
ejpam-976	134	11	.	.	PUNCT
ejpam-976	134	12	.	.	PUNCT
ejpam-976	134	13	.	.	PUNCT
ejpam-976	135	1	,	,	PUNCT
ejpam-976	135	2	n−	n−	NOUN
ejpam-976	135	3	1	1	NUM
ejpam-976	135	4	)	)	PUNCT
ejpam-976	135	5	represents	represent	VERB
ejpam-976	135	6	yearly	yearly	ADJ
ejpam-976	135	7	saving	saving	NOUN
ejpam-976	135	8	in	in	ADP
ejpam-976	135	9	period	period	NOUN
ejpam-976	135	10	i	i	PRON
ejpam-976	135	11	or	or	CCONJ
ejpam-976	135	12	amount	amount	NOUN
ejpam-976	135	13	invested	invest	VERB
ejpam-976	135	14	in	in	ADP
ejpam-976	135	15	period	period	NOUN
ejpam-976	135	16	i	i	PRON
ejpam-976	135	17	,	,	PUNCT
ejpam-976	135	18	yi+1	yi+1	PROPN
ejpam-976	135	19	refers	refer	VERB
ejpam-976	135	20	to	to	ADP
ejpam-976	135	21	the	the	DET
ejpam-976	135	22	random	random	ADJ
ejpam-976	135	23	rate	rate	NOUN
ejpam-976	135	24	of	of	ADP
ejpam-976	135	25	return	return	NOUN
ejpam-976	135	26	in	in	ADP
ejpam-976	135	27	period	period	NOUN
ejpam-976	135	28	i	i	PRON
ejpam-976	135	29	for	for	ADP
ejpam-976	135	30	i	i	PRON
ejpam-976	135	31	=	=	NOUN
ejpam-976	135	32	0	0	NUM
ejpam-976	135	33	,	,	PUNCT
ejpam-976	135	34	.	.	PUNCT
ejpam-976	135	35	.	.	PUNCT
ejpam-976	136	1	.	.	PUNCT
ejpam-976	137	1	,	,	PUNCT
ejpam-976	137	2	n−1	n−1	PROPN
ejpam-976	137	3	.	.	PUNCT
ejpam-976	138	1	the	the	DET
ejpam-976	138	2	term	term	NOUN
ejpam-976	138	3	yk	yk	PROPN
ejpam-976	138	4	=	=	NOUN
ejpam-976	138	5	log	log	PROPN
ejpam-976	138	6	pk	pk	PROPN
ejpam-976	138	7	pk−1	pk−1	PROPN
ejpam-976	138	8	=	=	PUNCT
ejpam-976	138	9	logpk−	logpk−	PROPN
ejpam-976	138	10	logpk−1	logpk−1	PROPN
ejpam-976	138	11	i.e	i.e	X
ejpam-976	138	12	eyk	eyk	NOUN
ejpam-976	138	13	=	=	SYM
ejpam-976	138	14	pk	pk	PROPN
ejpam-976	138	15	pk−1	pk−1	PROPN
ejpam-976	138	16	,	,	PUNCT
ejpam-976	138	17	where	where	SCONJ
ejpam-976	138	18	pk	pk	NOUN
ejpam-976	138	19	is	be	AUX
ejpam-976	138	20	the	the	DET
ejpam-976	138	21	price	price	NOUN
ejpam-976	138	22	of	of	ADP
ejpam-976	138	23	the	the	DET
ejpam-976	138	24	asset	asset	NOUN
ejpam-976	138	25	at	at	ADP
ejpam-976	138	26	period	period	NOUN
ejpam-976	138	27	k	k	PROPN
ejpam-976	138	28	,	,	PUNCT
ejpam-976	138	29	for	for	ADP
ejpam-976	138	30	k	k	PROPN
ejpam-976	138	31	=	=	SYM
ejpam-976	138	32	0	0	PROPN
ejpam-976	138	33	,	,	PUNCT
ejpam-976	138	34	.	.	PUNCT
ejpam-976	138	35	.	.	PUNCT
ejpam-976	139	1	.	.	PUNCT
ejpam-976	140	1	,	,	PUNCT
ejpam-976	140	2	n	n	CCONJ
ejpam-976	140	3	;	;	PUNCT
ejpam-976	140	4	is	be	AUX
ejpam-976	140	5	called	call	VERB
ejpam-976	140	6	the	the	DET
ejpam-976	140	7	random	random	ADJ
ejpam-976	140	8	log	log	NOUN
ejpam-976	140	9	-	-	PUNCT
ejpam-976	140	10	return	return	NOUN
ejpam-976	140	11	in	in	ADP
ejpam-976	140	12	period	period	NOUN
ejpam-976	140	13	k	k	PROPN
ejpam-976	140	14	and	and	CCONJ
ejpam-976	140	15	zi	zi	PROPN
ejpam-976	140	16	denote	denote	VERB
ejpam-976	140	17	the	the	DET
ejpam-976	140	18	sum	sum	NOUN
ejpam-976	140	19	of	of	ADP
ejpam-976	140	20	stochastic	stochastic	ADJ
ejpam-976	140	21	or	or	CCONJ
ejpam-976	140	22	random	random	ADJ
ejpam-976	140	23	returns	return	NOUN
ejpam-976	140	24	in	in	ADP
ejpam-976	140	25	period	period	NOUN
ejpam-976	140	26	i	i	PRON
ejpam-976	140	27	,	,	PUNCT
ejpam-976	140	28	i	i	PROPN
ejpam-976	140	29	=	=	NOUN
ejpam-976	140	30	0	0	NUM
ejpam-976	140	31	,	,	PUNCT
ejpam-976	140	32	.	.	PUNCT
ejpam-976	140	33	.	.	PUNCT
ejpam-976	141	1	.	.	PUNCT
ejpam-976	142	1	,	,	PUNCT
ejpam-976	142	2	n−	n−	NOUN
ejpam-976	142	3	1	1	NUM
ejpam-976	142	4	.	.	PUNCT
ejpam-976	142	5	with	with	ADP
ejpam-976	142	6	suitable	suitable	ADJ
ejpam-976	142	7	adjustment	adjustment	NOUN
ejpam-976	142	8	,	,	PUNCT
ejpam-976	142	9	s	s	PART
ejpam-976	142	10	could	could	AUX
ejpam-976	142	11	also	also	ADV
ejpam-976	142	12	be	be	AUX
ejpam-976	142	13	referred	refer	VERB
ejpam-976	142	14	as	as	ADP
ejpam-976	142	15	the	the	DET
ejpam-976	142	16	present	present	ADJ
ejpam-976	142	17	value	value	NOUN
ejpam-976	142	18	of	of	ADP
ejpam-976	142	19	a	a	DET
ejpam-976	142	20	series	series	NOUN
ejpam-976	142	21	of	of	ADP
ejpam-976	142	22	payments	payment	NOUN
ejpam-976	142	23	.	.	PUNCT
ejpam-976	143	1	to	to	PART
ejpam-976	143	2	be	be	AUX
ejpam-976	143	3	more	more	ADV
ejpam-976	143	4	precise	precise	ADJ
ejpam-976	143	5	,	,	PUNCT
ejpam-976	143	6	if	if	SCONJ
ejpam-976	143	7	−zi	−zi	PRON
ejpam-976	143	8	denotes	denote	VERB
ejpam-976	143	9	the	the	DET
ejpam-976	143	10	stochastic	stochastic	ADJ
ejpam-976	143	11	log	log	NOUN
ejpam-976	143	12	-	-	PUNCT
ejpam-976	143	13	return	return	NOUN
ejpam-976	143	14	over	over	ADP
ejpam-976	143	15	the	the	DET
ejpam-976	143	16	period	period	NOUN
ejpam-976	143	17	[	[	X
ejpam-976	143	18	0	0	NUM
ejpam-976	143	19	,	,	PUNCT
ejpam-976	143	20	i	i	PRON
ejpam-976	143	21	]	]	X
ejpam-976	143	22	,	,	PUNCT
ejpam-976	143	23	then	then	ADV
ejpam-976	143	24	ezi	ezi	PROPN
ejpam-976	143	25	,	,	PUNCT
ejpam-976	143	26	represents	represent	VERB
ejpam-976	143	27	the	the	DET
ejpam-976	143	28	stochastic	stochastic	ADJ
ejpam-976	143	29	discount	discount	NOUN
ejpam-976	143	30	factor	factor	NOUN
ejpam-976	143	31	over	over	ADP
ejpam-976	143	32	the	the	DET
ejpam-976	143	33	period	period	NOUN
ejpam-976	143	34	[	[	X
ejpam-976	143	35	0	0	NUM
ejpam-976	143	36	,	,	PUNCT
ejpam-976	143	37	i	i	PRON
ejpam-976	143	38	]	]	X
ejpam-976	143	39	.	.	PUNCT
ejpam-976	144	1	in	in	ADP
ejpam-976	144	2	this	this	DET
ejpam-976	144	3	situation	situation	NOUN
ejpam-976	144	4	,	,	PUNCT
ejpam-976	144	5	the	the	DET
ejpam-976	144	6	sum	sum	NOUN
ejpam-976	144	7	s	s	VERB
ejpam-976	144	8	is	be	AUX
ejpam-976	144	9	the	the	DET
ejpam-976	144	10	present	present	ADJ
ejpam-976	144	11	value	value	NOUN
ejpam-976	144	12	of	of	ADP
ejpam-976	144	13	αi	αi	PRON
ejpam-976	144	14	[	[	X
ejpam-976	144	15	4	4	NUM
ejpam-976	144	16	]	]	PUNCT
ejpam-976	144	17	.	.	PUNCT
ejpam-976	145	1	the	the	DET
ejpam-976	145	2	sum	sum	NOUN
ejpam-976	145	3	defined	define	VERB
ejpam-976	145	4	in	in	ADP
ejpam-976	145	5	(	(	PUNCT
ejpam-976	145	6	1	1	X
ejpam-976	145	7	)	)	PUNCT
ejpam-976	145	8	plays	play	VERB
ejpam-976	145	9	a	a	DET
ejpam-976	145	10	central	central	ADJ
ejpam-976	145	11	role	role	NOUN
ejpam-976	145	12	in	in	ADP
ejpam-976	145	13	the	the	DET
ejpam-976	145	14	actuarial	actuarial	ADJ
ejpam-976	145	15	and	and	CCONJ
ejpam-976	145	16	financial	financial	ADJ
ejpam-976	145	17	theory	theory	NOUN
ejpam-976	145	18	because	because	SCONJ
ejpam-976	145	19	it	it	PRON
ejpam-976	145	20	allows	allow	VERB
ejpam-976	145	21	computation	computation	NOUN
ejpam-976	145	22	of	of	ADP
ejpam-976	145	23	risk	risk	NOUN
ejpam-976	145	24	measures	measure	NOUN
ejpam-976	145	25	such	such	ADJ
ejpam-976	145	26	as	as	ADP
ejpam-976	145	27	value	value	NOUN
ejpam-976	145	28	at	at	ADP
ejpam-976	145	29	risk	risk	NOUN
ejpam-976	145	30	or	or	CCONJ
ejpam-976	145	31	stop	stop	VERB
ejpam-976	145	32	-	-	PUNCT
ejpam-976	145	33	loss	loss	NOUN
ejpam-976	145	34	premium	premium	NOUN
ejpam-976	145	35	.	.	PUNCT
ejpam-976	146	1	to	to	PART
ejpam-976	146	2	calculate	calculate	VERB
ejpam-976	146	3	the	the	DET
ejpam-976	146	4	risk	risk	NOUN
ejpam-976	146	5	measures	measure	NOUN
ejpam-976	146	6	we	we	PRON
ejpam-976	146	7	need	need	VERB
ejpam-976	146	8	to	to	PART
ejpam-976	146	9	evaluate	evaluate	VERB
ejpam-976	146	10	the	the	DET
ejpam-976	146	11	distribution	distribution	NOUN
ejpam-976	146	12	function	function	NOUN
ejpam-976	146	13	of	of	ADP
ejpam-976	146	14	s.	s.	PROPN
ejpam-976	146	15	unfortunately	unfortunately	ADV
ejpam-976	146	16	,	,	PUNCT
ejpam-976	146	17	the	the	DET
ejpam-976	146	18	distribution	distribution	NOUN
ejpam-976	146	19	of	of	ADP
ejpam-976	146	20	the	the	DET
ejpam-976	146	21	sum	sum	NOUN
ejpam-976	146	22	s	s	PART
ejpam-976	146	23	(	(	PUNCT
ejpam-976	146	24	of	of	ADP
ejpam-976	146	25	log	log	NOUN
ejpam-976	146	26	-	-	PUNCT
ejpam-976	146	27	normally	normally	ADV
ejpam-976	146	28	or	or	CCONJ
ejpam-976	146	29	log	log	NOUN
ejpam-976	146	30	-	-	PUNCT
ejpam-976	146	31	skew	skew	NOUN
ejpam-976	146	32	normally	normally	ADV
ejpam-976	146	33	distributed	distribute	VERB
ejpam-976	146	34	random	random	ADJ
ejpam-976	146	35	variables	variable	NOUN
ejpam-976	146	36	)	)	PUNCT
ejpam-976	146	37	is	be	AUX
ejpam-976	146	38	not	not	PART
ejpam-976	146	39	available	available	ADJ
ejpam-976	146	40	in	in	ADP
ejpam-976	146	41	the	the	DET
ejpam-976	146	42	closed	closed	ADJ
ejpam-976	146	43	-	-	PUNCT
ejpam-976	146	44	form	form	NOUN
ejpam-976	146	45	.	.	PUNCT
ejpam-976	147	1	it	it	PRON
ejpam-976	147	2	is	be	AUX
ejpam-976	147	3	possible	possible	ADJ
ejpam-976	147	4	to	to	PART
ejpam-976	147	5	use	use	VERB
ejpam-976	147	6	monte	monte	PROPN
ejpam-976	147	7	carlo	carlo	PROPN
ejpam-976	147	8	simulation	simulation	PROPN
ejpam-976	147	9	method	method	NOUN
ejpam-976	147	10	to	to	PART
ejpam-976	147	11	approximate	approximate	VERB
ejpam-976	147	12	the	the	DET
ejpam-976	147	13	distribution	distribution	NOUN
ejpam-976	147	14	function	function	NOUN
ejpam-976	147	15	.	.	PUNCT
ejpam-976	148	1	however	however	ADV
ejpam-976	148	2	,	,	PUNCT
ejpam-976	148	3	monte	monte	PROPN
ejpam-976	148	4	carlo	carlo	PROPN
ejpam-976	148	5	simulation	simulation	NOUN
ejpam-976	148	6	of	of	ADP
ejpam-976	148	7	the	the	DET
ejpam-976	148	8	distribution	distribution	NOUN
ejpam-976	148	9	is	be	AUX
ejpam-976	148	10	often	often	ADV
ejpam-976	148	11	time	time	NOUN
ejpam-976	148	12	-	-	PUNCT
ejpam-976	148	13	consuming	consume	VERB
ejpam-976	148	14	.	.	PUNCT
ejpam-976	149	1	thus	thus	ADV
ejpam-976	149	2	one	one	PRON
ejpam-976	149	3	has	have	VERB
ejpam-976	149	4	to	to	PART
ejpam-976	149	5	find	find	VERB
ejpam-976	149	6	alternate	alternate	ADJ
ejpam-976	149	7	way	way	NOUN
ejpam-976	149	8	to	to	PART
ejpam-976	149	9	approximate	approximate	VERB
ejpam-976	149	10	the	the	DET
ejpam-976	149	11	distribution	distribution	NOUN
ejpam-976	149	12	of	of	ADP
ejpam-976	149	13	the	the	DET
ejpam-976	149	14	sum	sum	NOUN
ejpam-976	149	15	.	.	PUNCT
ejpam-976	150	1	among	among	ADP
ejpam-976	150	2	the	the	DET
ejpam-976	150	3	proposed	propose	VERB
ejpam-976	150	4	solutions	solution	NOUN
ejpam-976	150	5	,	,	PUNCT
ejpam-976	150	6	moment	moment	NOUN
ejpam-976	150	7	matching	matching	NOUN
ejpam-976	150	8	methods	method	NOUN
ejpam-976	150	9	,	,	PUNCT
ejpam-976	150	10	lognormal	lognormal	ADJ
ejpam-976	150	11	and	and	CCONJ
ejpam-976	150	12	inverse	inverse	NOUN
ejpam-976	150	13	gamma	gamma	NOUN
ejpam-976	150	14	approximations	approximation	NOUN
ejpam-976	150	15	are	be	AUX
ejpam-976	150	16	commonly	commonly	ADV
ejpam-976	150	17	used	use	VERB
ejpam-976	150	18	.	.	PUNCT
ejpam-976	151	1	both	both	DET
ejpam-976	151	2	methods	method	NOUN
ejpam-976	151	3	approximate	approximate	VERB
ejpam-976	151	4	the	the	DET
ejpam-976	151	5	unknown	unknown	ADJ
ejpam-976	151	6	distribution	distribution	NOUN
ejpam-976	151	7	function	function	NOUN
ejpam-976	151	8	by	by	ADP
ejpam-976	151	9	a	a	DET
ejpam-976	151	10	given	give	VERB
ejpam-976	151	11	one	one	NUM
ejpam-976	151	12	such	such	ADJ
ejpam-976	151	13	that	that	SCONJ
ejpam-976	151	14	the	the	DET
ejpam-976	151	15	first	first	ADJ
ejpam-976	151	16	two	two	NUM
ejpam-976	151	17	moments	moment	NOUN
ejpam-976	151	18	coincide	coincide	NOUN
ejpam-976	151	19	.	.	PUNCT
ejpam-976	152	1	kaas	kaas	NOUN
ejpam-976	152	2	,	,	PUNCT
ejpam-976	152	3	et	et	PROPN
ejpam-976	152	4	al	al	PROPN
ejpam-976	152	5	.	.	PUNCT
ejpam-976	153	1	[	[	X
ejpam-976	153	2	2	2	NUM
ejpam-976	153	3	]	]	PUNCT
ejpam-976	153	4	and	and	CCONJ
ejpam-976	153	5	dhaene	dhaene	PROPN
ejpam-976	153	6	,	,	PUNCT
ejpam-976	153	7	et	et	PROPN
ejpam-976	153	8	al	al	PROPN
ejpam-976	153	9	.	.	PUNCT
ejpam-976	154	1	[	[	X
ejpam-976	154	2	1	1	X
ejpam-976	154	3	]	]	X
ejpam-976	154	4	propose	propose	VERB
ejpam-976	154	5	to	to	PART
ejpam-976	154	6	approximate	approximate	VERB
ejpam-976	154	7	the	the	DET
ejpam-976	154	8	distribution	distribution	NOUN
ejpam-976	154	9	function	function	NOUN
ejpam-976	154	10	references	reference	VERB
ejpam-976	154	11	784	784	NUM
ejpam-976	154	12	of	of	ADP
ejpam-976	154	13	s	s	PRON
ejpam-976	154	14	by	by	ADP
ejpam-976	154	15	so	so	ADV
ejpam-976	154	16	called	call	VERB
ejpam-976	154	17	“	"	PUNCT
ejpam-976	154	18	convex	convex	VERB
ejpam-976	154	19	lower	lower	ADV
ejpam-976	154	20	bound	bind	VERB
ejpam-976	154	21	”	"	PUNCT
ejpam-976	154	22	.	.	PUNCT
ejpam-976	155	1	the	the	DET
ejpam-976	155	2	underlying	underlying	ADJ
ejpam-976	155	3	idea	idea	NOUN
ejpam-976	155	4	of	of	ADP
ejpam-976	155	5	convex	convex	NOUN
ejpam-976	155	6	lower	low	ADJ
ejpam-976	155	7	order	order	NOUN
ejpam-976	155	8	bound	bind	VERB
ejpam-976	155	9	is	be	AUX
ejpam-976	155	10	to	to	PART
ejpam-976	155	11	replace	replace	VERB
ejpam-976	155	12	an	an	DET
ejpam-976	155	13	unknown	unknown	ADJ
ejpam-976	155	14	or	or	CCONJ
ejpam-976	155	15	too	too	ADV
ejpam-976	155	16	complex	complex	ADJ
ejpam-976	155	17	distribution	distribution	NOUN
ejpam-976	155	18	(	(	PUNCT
ejpam-976	155	19	for	for	ADP
ejpam-976	155	20	which	which	PRON
ejpam-976	155	21	no	no	DET
ejpam-976	155	22	explicit	explicit	ADJ
ejpam-976	155	23	form	form	NOUN
ejpam-976	155	24	is	be	AUX
ejpam-976	155	25	found	find	VERB
ejpam-976	155	26	)	)	PUNCT
ejpam-976	155	27	by	by	ADP
ejpam-976	155	28	another	another	DET
ejpam-976	155	29	one	one	NOUN
ejpam-976	155	30	which	which	PRON
ejpam-976	155	31	is	be	AUX
ejpam-976	155	32	easier	easy	ADJ
ejpam-976	155	33	to	to	PART
ejpam-976	155	34	determine	determine	VERB
ejpam-976	155	35	.	.	PUNCT
ejpam-976	156	1	in	in	ADP
ejpam-976	156	2	this	this	DET
ejpam-976	156	3	approach	approach	NOUN
ejpam-976	156	4	,	,	PUNCT
ejpam-976	156	5	the	the	DET
ejpam-976	156	6	real	real	ADJ
ejpam-976	156	7	distribution	distribution	NOUN
ejpam-976	156	8	is	be	AUX
ejpam-976	156	9	known	know	VERB
ejpam-976	156	10	to	to	PART
ejpam-976	156	11	be	be	AUX
ejpam-976	156	12	bounded	bound	VERB
ejpam-976	156	13	in	in	ADP
ejpam-976	156	14	terms	term	NOUN
ejpam-976	156	15	of	of	ADP
ejpam-976	156	16	convex	convex	NOUN
ejpam-976	156	17	ordering	order	VERB
ejpam-976	156	18	to	to	ADP
ejpam-976	156	19	the	the	DET
ejpam-976	156	20	approximated	approximate	VERB
ejpam-976	156	21	distribution	distribution	NOUN
ejpam-976	156	22	.	.	PUNCT
ejpam-976	157	1	to	to	PART
ejpam-976	157	2	be	be	AUX
ejpam-976	157	3	more	more	ADV
ejpam-976	157	4	precise	precise	ADJ
ejpam-976	157	5	,	,	PUNCT
ejpam-976	157	6	by	by	ADP
ejpam-976	157	7	theorem	theorem	NOUN
ejpam-976	157	8	1	1	NUM
ejpam-976	157	9	,	,	PUNCT
ejpam-976	157	10	the	the	DET
ejpam-976	157	11	distribution	distribution	NOUN
ejpam-976	157	12	function	function	NOUN
ejpam-976	157	13	of	of	ADP
ejpam-976	157	14	s	s	NOUN
ejpam-976	157	15	=	=	PUNCT
ejpam-976	157	16	∑n−1	∑n−1	ADJ
ejpam-976	158	1	i=0	i=0	PROPN
ejpam-976	158	2	αie	αie	PROPN
ejpam-976	158	3	zi	zi	PROPN
ejpam-976	158	4	is	be	AUX
ejpam-976	158	5	approximated	approximate	VERB
ejpam-976	158	6	by	by	ADP
ejpam-976	158	7	the	the	DET
ejpam-976	158	8	distribution	distribution	NOUN
ejpam-976	158	9	function	function	NOUN
ejpam-976	158	10	of	of	ADP
ejpam-976	158	11	sl	sl	INTJ
ejpam-976	158	12	,	,	PUNCT
ejpam-976	158	13	where	where	SCONJ
ejpam-976	158	14	sl	sl	PROPN
ejpam-976	158	15	is	be	AUX
ejpam-976	158	16	defined	define	VERB
ejpam-976	158	17	by	by	ADP
ejpam-976	158	18	,	,	PUNCT
ejpam-976	158	19	sl	sl	PROPN
ejpam-976	158	20	=	=	VERB
ejpam-976	158	21	n−1∑	n−1∑	PROPN
ejpam-976	158	22	i=0	i=0	PROPN
ejpam-976	158	23	αi	αi	PART
ejpam-976	158	24	e(e	e(e	NOUN
ejpam-976	158	25	zi	zi	PROPN
ejpam-976	158	26	|λ	|λ	PROPN
ejpam-976	158	27	)	)	PUNCT
ejpam-976	158	28	(	(	PUNCT
ejpam-976	158	29	2	2	X
ejpam-976	158	30	)	)	PUNCT
ejpam-976	158	31	an	an	DET
ejpam-976	158	32	appropriate	appropriate	ADJ
ejpam-976	158	33	choice	choice	NOUN
ejpam-976	158	34	of	of	ADP
ejpam-976	158	35	the	the	DET
ejpam-976	158	36	conditioning	conditioning	NOUN
ejpam-976	158	37	random	random	ADJ
ejpam-976	158	38	variable	variable	NOUN
ejpam-976	158	39	λ	λ	NOUN
ejpam-976	158	40	is	be	AUX
ejpam-976	158	41	required	require	VERB
ejpam-976	158	42	.	.	PUNCT
ejpam-976	159	1	this	this	DET
ejpam-976	159	2	approach	approach	NOUN
ejpam-976	159	3	has	have	VERB
ejpam-976	159	4	two	two	NUM
ejpam-976	159	5	-	-	PUNCT
ejpam-976	159	6	fold	fold	ADJ
ejpam-976	159	7	advantages	advantage	NOUN
ejpam-976	159	8	.	.	PUNCT
ejpam-976	160	1	firstly	firstly	ADV
ejpam-976	160	2	,	,	PUNCT
ejpam-976	160	3	use	use	NOUN
ejpam-976	160	4	of	of	ADP
ejpam-976	160	5	this	this	DET
ejpam-976	160	6	approach	approach	NOUN
ejpam-976	160	7	transforms	transform	VERB
ejpam-976	160	8	the	the	DET
ejpam-976	160	9	multidimensionality	multidimensionality	NOUN
ejpam-976	160	10	problem	problem	NOUN
ejpam-976	160	11	caused	cause	VERB
ejpam-976	160	12	by	by	ADP
ejpam-976	160	13	(	(	PUNCT
ejpam-976	160	14	z0	z0	PROPN
ejpam-976	160	15	,	,	PUNCT
ejpam-976	160	16	z2	z2	PROPN
ejpam-976	160	17	,	,	PUNCT
ejpam-976	160	18	.	.	PUNCT
ejpam-976	160	19	.	.	PUNCT
ejpam-976	161	1	.	.	PUNCT
ejpam-976	162	1	,	,	PUNCT
ejpam-976	162	2	zn−1	zn−1	PROPN
ejpam-976	162	3	)	)	PUNCT
ejpam-976	162	4	to	to	ADP
ejpam-976	162	5	a	a	DET
ejpam-976	162	6	single	single	ADJ
ejpam-976	162	7	dimension	dimension	NOUN
ejpam-976	162	8	caused	cause	VERB
ejpam-976	162	9	by	by	ADP
ejpam-976	162	10	λ	λ	NOUN
ejpam-976	162	11	.	.	PUNCT
ejpam-976	162	12	secondly	secondly	ADV
ejpam-976	162	13	,	,	PUNCT
ejpam-976	162	14	an	an	DET
ejpam-976	162	15	appropriate	appropriate	ADJ
ejpam-976	162	16	choice	choice	NOUN
ejpam-976	162	17	of	of	ADP
ejpam-976	162	18	λ	λ	PROPN
ejpam-976	162	19	(	(	PUNCT
ejpam-976	162	20	that	that	PRON
ejpam-976	162	21	makes	make	VERB
ejpam-976	162	22	the	the	DET
ejpam-976	162	23	expectation	expectation	NOUN
ejpam-976	162	24	in	in	ADP
ejpam-976	162	25	(	(	PUNCT
ejpam-976	162	26	2	2	NUM
ejpam-976	162	27	)	)	PUNCT
ejpam-976	162	28	non	non	ADJ
ejpam-976	162	29	-	-	ADJ
ejpam-976	162	30	decreasing	decrease	VERB
ejpam-976	162	31	or	or	CCONJ
ejpam-976	162	32	non	non	ADJ
ejpam-976	162	33	-	-	ADJ
ejpam-976	162	34	increasing	increasing	ADJ
ejpam-976	162	35	function	function	NOUN
ejpam-976	162	36	of	of	ADP
ejpam-976	162	37	the	the	DET
ejpam-976	162	38	conditioning	conditioning	NOUN
ejpam-976	162	39	random	random	ADJ
ejpam-976	162	40	variable	variable	ADJ
ejpam-976	162	41	λ	λ	NOUN
ejpam-976	162	42	)	)	PUNCT
ejpam-976	162	43	will	will	AUX
ejpam-976	162	44	make	make	VERB
ejpam-976	162	45	a	a	DET
ejpam-976	162	46	comonotonic	comonotonic	ADJ
ejpam-976	162	47	sum	sum	NOUN
ejpam-976	162	48	i.e	i.e	PRON
ejpam-976	162	49	the	the	DET
ejpam-976	162	50	elements	element	NOUN
ejpam-976	162	51	of	of	ADP
ejpam-976	162	52	the	the	DET
ejpam-976	162	53	sum	sum	NOUN
ejpam-976	162	54	in	in	ADP
ejpam-976	162	55	(	(	PUNCT
ejpam-976	162	56	2	2	X
ejpam-976	162	57	)	)	PUNCT
ejpam-976	162	58	posses	posse	NOUN
ejpam-976	162	59	the	the	DET
ejpam-976	162	60	so	so	ADV
ejpam-976	162	61	called	call	VERB
ejpam-976	162	62	comonotonic	comonotonic	ADJ
ejpam-976	162	63	dependence	dependence	NOUN
ejpam-976	162	64	structure	structure	NOUN
ejpam-976	162	65	.	.	PUNCT
ejpam-976	163	1	using	use	VERB
ejpam-976	163	2	additivity	additivity	NOUN
ejpam-976	163	3	properties	property	NOUN
ejpam-976	163	4	of	of	ADP
ejpam-976	163	5	sum	sum	NOUN
ejpam-976	163	6	of	of	ADP
ejpam-976	163	7	comonotonic	comonotonic	ADJ
ejpam-976	163	8	random	random	ADJ
ejpam-976	163	9	variables	variable	NOUN
ejpam-976	163	10	risk	risk	NOUN
ejpam-976	163	11	measures	measure	NOUN
ejpam-976	163	12	related	relate	VERB
ejpam-976	163	13	to	to	ADP
ejpam-976	163	14	the	the	DET
ejpam-976	163	15	distribution	distribution	NOUN
ejpam-976	163	16	function	function	NOUN
ejpam-976	163	17	of	of	ADP
ejpam-976	163	18	s	s	PROPN
ejpam-976	163	19	is	be	AUX
ejpam-976	163	20	then	then	ADV
ejpam-976	163	21	approximated	approximate	VERB
ejpam-976	163	22	by	by	ADP
ejpam-976	163	23	the	the	DET
ejpam-976	163	24	corresponding	corresponding	ADJ
ejpam-976	163	25	risk	risk	NOUN
ejpam-976	163	26	measures	measure	NOUN
ejpam-976	163	27	of	of	ADP
ejpam-976	163	28	sl	sl	NOUN
ejpam-976	163	29	.	.	PUNCT
ejpam-976	164	1	according	accord	VERB
ejpam-976	164	2	to	to	ADP
ejpam-976	164	3	[	[	X
ejpam-976	164	4	2	2	NUM
ejpam-976	164	5	]	]	PUNCT
ejpam-976	164	6	,	,	PUNCT
ejpam-976	164	7	comonotonic	comonotonic	ADJ
ejpam-976	164	8	upper	upper	ADJ
ejpam-976	164	9	bound	bind	VERB
ejpam-976	164	10	for	for	ADP
ejpam-976	164	11	the	the	DET
ejpam-976	164	12	sum	sum	NOUN
ejpam-976	164	13	in	in	ADP
ejpam-976	164	14	convex	convex	ADJ
ejpam-976	164	15	order	order	NOUN
ejpam-976	164	16	sense	sense	NOUN
ejpam-976	164	17	can	can	AUX
ejpam-976	164	18	also	also	ADV
ejpam-976	164	19	be	be	AUX
ejpam-976	164	20	derived	derive	VERB
ejpam-976	164	21	using	use	VERB
ejpam-976	164	22	the	the	DET
ejpam-976	164	23	result	result	NOUN
ejpam-976	164	24	n−1∑	n−1∑	PROPN
ejpam-976	164	25	i=0	i=0	PROPN
ejpam-976	164	26	x	x	PUNCT
ejpam-976	164	27	i	i	PRON
ejpam-976	164	28	≤cx	≤cx	VERB
ejpam-976	164	29	n−1∑	n−1∑	PROPN
ejpam-976	164	30	i=0	i=0	PROPN
ejpam-976	164	31	fx	fx	NOUN
ejpam-976	165	1	i	i	PRON
ejpam-976	165	2	(	(	PUNCT
ejpam-976	165	3	u	u	NOUN
ejpam-976	165	4	)	)	PUNCT
ejpam-976	165	5	,	,	PUNCT
ejpam-976	165	6	where	where	SCONJ
ejpam-976	165	7	u	u	NOUN
ejpam-976	165	8	is	be	AUX
ejpam-976	165	9	the	the	DET
ejpam-976	165	10	uniform	uniform	ADJ
ejpam-976	165	11	random	random	ADJ
ejpam-976	165	12	variable	variable	NOUN
ejpam-976	165	13	over	over	ADP
ejpam-976	165	14	(	(	PUNCT
ejpam-976	165	15	0,1	0,1	NUM
ejpam-976	165	16	)	)	PUNCT
ejpam-976	165	17	.	.	PUNCT
ejpam-976	166	1	however	however	ADV
ejpam-976	166	2	,	,	PUNCT
ejpam-976	166	3	comonotonic	comonotonic	ADJ
ejpam-976	166	4	upper	upper	ADJ
ejpam-976	166	5	bounds	bound	NOUN
ejpam-976	166	6	generally	generally	ADV
ejpam-976	166	7	provide	provide	VERB
ejpam-976	166	8	too	too	ADV
ejpam-976	166	9	conservative	conservative	ADJ
ejpam-976	166	10	estimates	estimate	NOUN
ejpam-976	166	11	of	of	ADP
ejpam-976	166	12	the	the	DET
ejpam-976	166	13	cumulative	cumulative	ADJ
ejpam-976	166	14	distribution	distribution	NOUN
ejpam-976	166	15	function	function	NOUN
ejpam-976	166	16	[	[	X
ejpam-976	166	17	3	3	NUM
ejpam-976	166	18	]	]	PUNCT
ejpam-976	166	19	.	.	PUNCT
ejpam-976	167	1	thus	thus	ADV
ejpam-976	167	2	we	we	PRON
ejpam-976	167	3	only	only	ADV
ejpam-976	167	4	discussed	discuss	VERB
ejpam-976	167	5	convex	convex	NOUN
ejpam-976	167	6	lower	lower	ADV
ejpam-976	167	7	bound	bind	VERB
ejpam-976	167	8	here	here	ADV
ejpam-976	167	9	.	.	PUNCT
ejpam-976	168	1	remark	remark	VERB
ejpam-976	168	2	1	1	NUM
ejpam-976	168	3	.	.	PUNCT
ejpam-976	169	1	in	in	ADP
ejpam-976	169	2	general	general	ADJ
ejpam-976	169	3	,	,	PUNCT
ejpam-976	169	4	the	the	DET
ejpam-976	169	5	random	random	ADJ
ejpam-976	169	6	vector	vector	NOUN
ejpam-976	169	7	(	(	PUNCT
ejpam-976	169	8	e(x0|λ	e(x0|λ	PROPN
ejpam-976	169	9	)	)	PUNCT
ejpam-976	169	10	,	,	PUNCT
ejpam-976	169	11	e(x1|λ	e(x1|λ	PROPN
ejpam-976	169	12	)	)	PUNCT
ejpam-976	169	13	,	,	PUNCT
ejpam-976	169	14	.	.	PUNCT
ejpam-976	169	15	.	.	PUNCT
ejpam-976	169	16	.	.	PUNCT
ejpam-976	170	1	,	,	PUNCT
ejpam-976	170	2	e(xn−1|λ	e(xn−1|λ	NOUN
ejpam-976	170	3	)	)	PUNCT
ejpam-976	170	4	)	)	PUNCT
ejpam-976	170	5	does	do	AUX
ejpam-976	170	6	not	not	PART
ejpam-976	170	7	have	have	VERB
ejpam-976	170	8	the	the	DET
ejpam-976	170	9	same	same	ADJ
ejpam-976	170	10	marginal	marginal	ADJ
ejpam-976	170	11	distribution	distribution	NOUN
ejpam-976	170	12	as	as	ADP
ejpam-976	170	13	(	(	PUNCT
ejpam-976	170	14	x0	x0	PROPN
ejpam-976	170	15	,	,	PUNCT
ejpam-976	170	16	x1	x1	PROPN
ejpam-976	170	17	,	,	PUNCT
ejpam-976	170	18	.	.	PUNCT
ejpam-976	170	19	.	.	PUNCT
ejpam-976	171	1	.	.	PUNCT
ejpam-976	172	1	,	,	PUNCT
ejpam-976	172	2	xn−1	xn−1	PROPN
ejpam-976	172	3	)	)	PUNCT
ejpam-976	172	4	.	.	PUNCT
ejpam-976	173	1	however	however	ADV
ejpam-976	173	2	,	,	PUNCT
ejpam-976	173	3	if	if	SCONJ
ejpam-976	173	4	the	the	DET
ejpam-976	173	5	conditioning	condition	VERB
ejpam-976	173	6	random	random	ADJ
ejpam-976	173	7	variable	variable	ADJ
ejpam-976	173	8	λ	λ	NOUN
ejpam-976	173	9	is	be	AUX
ejpam-976	173	10	chosen	choose	VERB
ejpam-976	173	11	in	in	ADP
ejpam-976	173	12	such	such	DET
ejpam-976	173	13	a	a	DET
ejpam-976	173	14	way	way	NOUN
ejpam-976	173	15	that	that	PRON
ejpam-976	173	16	all	all	DET
ejpam-976	173	17	random	random	ADJ
ejpam-976	173	18	variables	variable	NOUN
ejpam-976	173	19	e(x	e(x	NUM
ejpam-976	173	20	i|λ	i|λ	NOUN
ejpam-976	173	21	)	)	PUNCT
ejpam-976	173	22	,	,	PUNCT
ejpam-976	173	23	(	(	PUNCT
ejpam-976	173	24	i	i	NOUN
ejpam-976	173	25	=	=	NOUN
ejpam-976	173	26	0,1,2	0,1,2	NUM
ejpam-976	173	27	,	,	PUNCT
ejpam-976	173	28	.	.	PUNCT
ejpam-976	173	29	.	.	PUNCT
ejpam-976	173	30	.	.	PUNCT
ejpam-976	174	1	,	,	PUNCT
ejpam-976	174	2	n−	n−	NOUN
ejpam-976	174	3	1	1	NUM
ejpam-976	174	4	)	)	PUNCT
ejpam-976	174	5	are	be	AUX
ejpam-976	174	6	nondecreasing	nondecrease	VERB
ejpam-976	174	7	functions	function	NOUN
ejpam-976	174	8	of	of	ADP
ejpam-976	174	9	λ	λ	PROPN
ejpam-976	174	10	(	(	PUNCT
ejpam-976	174	11	or	or	CCONJ
ejpam-976	174	12	non	non	ADJ
ejpam-976	174	13	-	-	ADJ
ejpam-976	174	14	increasing	increasing	ADJ
ejpam-976	174	15	functions	function	NOUN
ejpam-976	174	16	of	of	ADP
ejpam-976	174	17	λ	λ	NOUN
ejpam-976	174	18	)	)	PUNCT
ejpam-976	174	19	,	,	PUNCT
ejpam-976	174	20	then	then	ADV
ejpam-976	174	21	the	the	DET
ejpam-976	174	22	sum	sum	NOUN
ejpam-976	174	23	∑n−1	∑n−1	ADP
ejpam-976	174	24	i=0	i=0	ADJ
ejpam-976	174	25	e[x	e[x	NUM
ejpam-976	174	26	i|λ	i|λ	NOUN
ejpam-976	174	27	]	]	PUNCT
ejpam-976	174	28	is	be	AUX
ejpam-976	174	29	a	a	DET
ejpam-976	174	30	sum	sum	NOUN
ejpam-976	174	31	of	of	ADP
ejpam-976	174	32	n	n	CCONJ
ejpam-976	174	33	comonotonous	comonotonous	ADJ
ejpam-976	174	34	random	random	ADJ
ejpam-976	174	35	variables	variable	NOUN
ejpam-976	174	36	and	and	CCONJ
ejpam-976	174	37	can	can	AUX
ejpam-976	174	38	be	be	AUX
ejpam-976	174	39	referred	refer	VERB
ejpam-976	174	40	to	to	ADP
ejpam-976	174	41	as	as	ADP
ejpam-976	174	42	comonotonic	comonotonic	X
ejpam-976	174	43	lower	lower	ADV
ejpam-976	174	44	bound	bind	VERB
ejpam-976	174	45	.	.	PUNCT
ejpam-976	175	1	hence	hence	ADV
ejpam-976	175	2	risk	risk	NOUN
ejpam-976	175	3	measures	measure	NOUN
ejpam-976	175	4	for	for	ADP
ejpam-976	175	5	the	the	DET
ejpam-976	175	6	sum	sum	NOUN
ejpam-976	175	7	could	could	AUX
ejpam-976	175	8	easily	easily	ADV
ejpam-976	175	9	be	be	AUX
ejpam-976	175	10	obtained	obtain	VERB
ejpam-976	175	11	by	by	ADP
ejpam-976	175	12	summing	sum	VERB
ejpam-976	175	13	the	the	DET
ejpam-976	175	14	corresponding	correspond	VERB
ejpam-976	175	15	risk	risk	NOUN
ejpam-976	175	16	measures	measure	NOUN
ejpam-976	175	17	for	for	ADP
ejpam-976	175	18	the	the	DET
ejpam-976	175	19	marginals	marginal	NOUN
ejpam-976	175	20	involved	involve	VERB
ejpam-976	175	21	.	.	PUNCT
ejpam-976	176	1	references	reference	NOUN
ejpam-976	176	2	[	[	X
ejpam-976	176	3	1	1	NUM
ejpam-976	176	4	]	]	PUNCT
ejpam-976	176	5	j	j	PROPN
ejpam-976	176	6	dhaene	dhaene	PROPN
ejpam-976	176	7	,	,	PUNCT
ejpam-976	176	8	m	m	VERB
ejpam-976	176	9	denuit	denuit	ADJ
ejpam-976	176	10	,	,	PUNCT
ejpam-976	176	11	m	m	NOUN
ejpam-976	176	12	gooverts	goovert	NOUN
ejpam-976	176	13	,	,	PUNCT
ejpam-976	176	14	r	r	NOUN
ejpam-976	176	15	kass	kass	NOUN
ejpam-976	176	16	,	,	PUNCT
ejpam-976	176	17	and	and	CCONJ
ejpam-976	176	18	d	d	ADP
ejpam-976	176	19	vyncke	vyncke	NOUN
ejpam-976	176	20	.	.	PUNCT
ejpam-976	177	1	the	the	DET
ejpam-976	177	2	concept	concept	NOUN
ejpam-976	177	3	of	of	ADP
ejpam-976	177	4	comonotonicity	comonotonicity	NOUN
ejpam-976	177	5	in	in	ADP
ejpam-976	177	6	actuarial	actuarial	ADJ
ejpam-976	177	7	science	science	NOUN
ejpam-976	177	8	and	and	CCONJ
ejpam-976	177	9	finance	finance	NOUN
ejpam-976	177	10	:	:	PUNCT
ejpam-976	177	11	applications	application	NOUN
ejpam-976	177	12	.	.	PUNCT
ejpam-976	178	1	insurance	insurance	NOUN
ejpam-976	178	2	:	:	PUNCT
ejpam-976	178	3	mathematics	mathematic	NOUN
ejpam-976	178	4	and	and	CCONJ
ejpam-976	178	5	economics	economic	NOUN
ejpam-976	178	6	,	,	PUNCT
ejpam-976	178	7	31:133–161	31:133–161	NUM
ejpam-976	178	8	,	,	PUNCT
ejpam-976	178	9	2002	2002	NUM
ejpam-976	178	10	.	.	PUNCT
ejpam-976	179	1	[	[	X
ejpam-976	179	2	2	2	NUM
ejpam-976	179	3	]	]	PUNCT
ejpam-976	179	4	j	j	PROPN
ejpam-976	179	5	dhaene	dhaene	PROPN
ejpam-976	179	6	r	r	PROPN
ejpam-976	179	7	kaas	kaas	NOUN
ejpam-976	179	8	,	,	PUNCT
ejpam-976	179	9	m	m	NOUN
ejpam-976	179	10	goovaerts	goovaert	NOUN
ejpam-976	179	11	and	and	CCONJ
ejpam-976	179	12	m	m	VERB
ejpam-976	179	13	denuit	denuit	ADJ
ejpam-976	179	14	.	.	PUNCT
ejpam-976	180	1	modern	modern	ADJ
ejpam-976	180	2	actuarial	actuarial	ADJ
ejpam-976	180	3	risk	risk	NOUN
ejpam-976	180	4	theory	theory	NOUN
ejpam-976	180	5	.	.	PUNCT
ejpam-976	181	1	kluwer	kluwer	NOUN
ejpam-976	181	2	academic	academic	ADJ
ejpam-976	181	3	publishers	publisher	NOUN
ejpam-976	181	4	,	,	PUNCT
ejpam-976	181	5	dordrecht	dordrecht	PROPN
ejpam-976	181	6	,	,	PUNCT
ejpam-976	181	7	2001	2001	NUM
ejpam-976	181	8	.	.	PUNCT
ejpam-976	182	1	references	reference	NOUN
ejpam-976	182	2	785	785	NUM
ejpam-976	183	1	[	[	X
ejpam-976	183	2	3	3	NUM
ejpam-976	183	3	]	]	X
ejpam-976	183	4	o	o	X
ejpam-976	183	5	roch	roch	PROPN
ejpam-976	183	6	and	and	CCONJ
ejpam-976	183	7	e	e	PROPN
ejpam-976	183	8	valdez	valdez	PROPN
ejpam-976	183	9	.	.	PUNCT
ejpam-976	184	1	lower	low	ADJ
ejpam-976	184	2	convex	convex	ADJ
ejpam-976	184	3	order	order	NOUN
ejpam-976	184	4	bound	bind	VERB
ejpam-976	184	5	approximations	approximation	NOUN
ejpam-976	184	6	for	for	ADP
ejpam-976	184	7	sums	sum	NOUN
ejpam-976	184	8	of	of	ADP
ejpam-976	184	9	log	log	NOUN
ejpam-976	184	10	-	-	PUNCT
ejpam-976	184	11	skew	skew	NOUN
ejpam-976	184	12	normal	normal	ADJ
ejpam-976	184	13	random	random	ADJ
ejpam-976	184	14	variables	variable	NOUN
ejpam-976	184	15	.	.	PUNCT
ejpam-976	185	1	working	work	VERB
ejpam-976	185	2	paper	paper	NOUN
ejpam-976	185	3	(	(	PUNCT
ejpam-976	185	4	2008	2008	NUM
ejpam-976	185	5	)	)	PUNCT
ejpam-976	185	6	.	.	PUNCT
ejpam-976	186	1	[	[	X
ejpam-976	186	2	4	4	NUM
ejpam-976	186	3	]	]	PUNCT
ejpam-976	186	4	s	s	PART
ejpam-976	186	5	vanduffel	vanduffel	NOUN
ejpam-976	186	6	,	,	PUNCT
ejpam-976	186	7	x	x	PROPN
ejpam-976	186	8	chen	chen	PROPN
ejpam-976	186	9	,	,	PUNCT
ejpam-976	186	10	j	j	PROPN
ejpam-976	186	11	dhaene	dhaene	PROPN
ejpam-976	186	12	,	,	PUNCT
ejpam-976	186	13	m	m	VERB
ejpam-976	186	14	goovaerts	goovaert	NOUN
ejpam-976	186	15	,	,	PUNCT
ejpam-976	186	16	l	l	PROPN
ejpam-976	186	17	henrard	henrard	NOUN
ejpam-976	186	18	,	,	PUNCT
ejpam-976	186	19	and	and	CCONJ
ejpam-976	186	20	r	r	NOUN
ejpam-976	186	21	kass	kass	NOUN
ejpam-976	186	22	.	.	PUNCT
ejpam-976	186	23	optimal	optimal	ADJ
ejpam-976	186	24	approximations	approximation	NOUN
ejpam-976	186	25	for	for	ADP
ejpam-976	186	26	risk	risk	NOUN
ejpam-976	186	27	measures	measure	NOUN
ejpam-976	186	28	of	of	ADP
ejpam-976	186	29	sums	sum	NOUN
ejpam-976	186	30	of	of	ADP
ejpam-976	186	31	lognormals	lognormal	NOUN
ejpam-976	186	32	based	base	VERB
ejpam-976	186	33	on	on	ADP
ejpam-976	186	34	conditional	conditional	ADJ
ejpam-976	186	35	expectations	expectation	NOUN
ejpam-976	186	36	.	.	PUNCT
ejpam-976	187	1	journal	journal	NOUN
ejpam-976	187	2	of	of	ADP
ejpam-976	187	3	computational	computational	ADJ
ejpam-976	187	4	and	and	CCONJ
ejpam-976	187	5	applied	applied	ADJ
ejpam-976	187	6	mathematics	mathematic	NOUN
ejpam-976	187	7	,	,	PUNCT
ejpam-976	187	8	221:202–218	221:202–218	NUM
ejpam-976	187	9	,	,	PUNCT
ejpam-976	187	10	2008	2008	NUM
ejpam-976	187	11	.	.	PUNCT