Subexponential Tails of Discounted Aggregate Claimsin a Time-Dependent Renewal Risk Model

Jinzhu Li[a];[b], Qihe Tang[b];�, and Rong Wu[a][a] School of Mathematical Science and LPMC

Nankai University, Tianjin 300071, P.R. China

E-mails: lijinzhunk@gmail.com (J. Li); rongwu@nankai.edu.cn (R. Wu)[b] Department of Statistics and Actuarial Science, The University of Iowa

241 Schae¤er Hall, Iowa City, IA 52242, USA

E-mail: qtang@stat.uiowa.edu

August 23, 2010

Abstract

Consider a continuous-time renewal risk model with a constant force of interest.We assume that claim sizes and inter-arrival times correspondingly form a sequenceof independent and identically distributed random pairs and that each pair obeys adependence structure described via the conditional tail probability of a claim size giventhe inter-arrival time before the claim. We focus on determination of the impact ofthis dependence structure on the asymptotic tail probability of discounted aggregateclaims. Assuming that the claim-size distribution is subexponential, we derive an exactlocally uniform asymptotic formula, which quantitatively captures the impact of thedependence structure. When the claim-size distribution is extended-regularly-varyingtailed, we show that this asymptotic formula is globally uniform.Keywords: Asymptotics; Dependence; Discounted aggregate claims; Extended-

regular variation; Subexponentiality; Uniformity.Mathematics Subject Classi�cation: Primary 62P05; Secondary 62H20, 60E05

1 Introduction

The renewal risk model has been playing a fundamental role in classical and modern risktheory since it was introduced by Sparre Andersen in the middle of last century as a naturalgeneralization of the compound Poisson risk model. In the standard framework of the renewalrisk model, both claim sizes Xk, k = 1; 2; : : :, and inter-arrival times �k, k = 1; 2; : : :, forma sequence of independent and identically distributed (i.i.d.) random variables and the twosequences are mutually independent too. Denote by X and � the generic random variables

�Corresponding author: Qihe Tang; Tel.: 1-319-335�0730; Fax: 1-319-335-3017.

1

of the claim sizes and inter-arrival times, respectively. Being equipped with other modelingfactors (such as initial surplus, premium incomes) and incorporated with some economicfactors (such as interests, dividends, taxes, returns on investments), this model provides agood mechanism for describing non-life insurance business.It should be noted that these independence assumptions are made mainly not for prac-

tical relevance but for mathematical tractability. Among them, the one assuming completeindependence between the claim size X and the inter-arrival time � is especially unrealisticin almost all kinds of insurance. Think that, if the deductible retained to insureds is raised,then the inter-arrival time will increase because small claims will be ruled out, while thelikelihood of a large claim will increase if X is new-worse-than-used or decrease if X is new-better-than-used. However, it is usually challenging to get results without such independenceassumptions because most of available methods will fail to work.During the last few years things have started to change. Especially in ruin theory, various

non-standard extensions to the renewal risk model have been proposed to appropriatelyrelax these independence assumptions. Initiated by Albrecher and Teugels (2006), therehave been a series of papers devoted to ruin-related problems of an extended renewal riskmodel in which (Xk; �k), k = 1; 2; : : :, are assumed to be i.i.d. copies of a generic pair (X; �)with dependent components X and �. An advantage of this risk model is that independencebetween the increments of the surplus process over claim arrival times is preserved. Albrecherand Teugels (2006) considered that (X; �) follows an arbitrary dependence structure, througha copula, and they derived explicit exponential estimates for �nite-time and in�nite-timeruin probabilities in the case of light-tailed claim sizes. Boudreault et al. (2006) proposed adependence structure thatX conditional on � has a density function equal to a mixture of twoarbitrary density functions, and they studied the Gerber�Shiu expected discounted penaltyfunction and measured the impact of the dependence structure on the ruin probability viathe comparison of Lundberg coe¢ cients. Cossette et al. (2008) assumed a generalized Farlie�Gumbel�Morgenstern copula to describe the dependence structure of (X; �), and they derivedthe Laplace transform of the Gerber�Shiu function. Badescu et al. (2009) assumed that(X; �) follows a bivariate phase-type distribution, and they employed the existing connectionbetween risk processes and �uid �ows to the analysis of various ruin-related quantities. Veryrecently, Asimit and Badescu (2010) introduced a general dependence structure for (X; �), viathe conditional tail probability of X given �, and they studied the tail behavior of discountedaggregate claims in the compound Poisson risk model in the presence of a constant force ofinterest and heavy-tailed claim sizes.Other non-standard extensions to the renewal risk model, not within the above-mentioned

framework, can also be found in Asmussen et al. (1999), Albrecher and Boxma (2004, 2005),and Biard et al. (2008), among others.In this paper we use the same dependence structure as proposed by Asimit and Badescu

(2010) for the generic random pair (X; �). Precisely, we assume that the claim size X andthe inter-arrival time � ful�ll the relation

Pr (X > xj� = t) � Pr (X > x)h(t); t � 0; (1.1)

2

for some measurable function h(�) : [0;1) 7�! (0;1), where the symbol � means that thequotient of both sides tends to 1 as x ! 1. When t is not a possible value of �, that is,Pr (� 2 �) = 0 for some open interval � containing t, the conditional probability in (1.1) issimply understood as unconditional and, therefore, h(t) = 1. Actually, in our main resultsbelow, whenever h(t) appears it is multiplied by Pr(� 2 dt). Hence, the function h(t) at sucha point t can be assigned any positive value without a¤ecting our �nal results. As discussedby Asimit and Badescu (2010) (see also Section 3 below), relation (1.1) de�nes a generaldependence structure which is easily veri�able for some commonly-used bivariate copulasand allows both positive and negative dependence. It also brings us great convenience indealing with the tail behavior of the sum or product of two dependent random variables. Forinstance, consider the discounted value Xe�r� with r � 0. If relation (1.1) holds uniformlyfor t 2 [0;1), which is often the case in concrete examples, then integrating both sides withrespect to Pr(� 2 dt) leads to Eh(�) = 1. Hence, by conditioning on � we obtain

Pr�Xe�r� > x

��Z 1

0�Pr�X > xert

�h(t) Pr(� 2 dt) = Pr

�Xe�r�

�> x

�; (1.2)

where �� is a random variable, independent of X, with a proper distribution given by

Pr(�� 2 dt) = h(t) Pr(� 2 dt):

The analysis in relation (1.2) shows that the dependence structure de�ned by (1.1) can beeasily dissolved and its impact on the tail behavior of quantities under consideration can beeasily captured.Consider the renewal risk model in which (Xk; �k), k = 1; 2; : : :, are i.i.d. copies of

a generic pair (X; �) ful�lling the dependence structure de�ned by (1.1). In this paper,assuming a constant force of interest r � 0 and heavy-tailed claim sizes, we study the tailbehavior of discounted aggregate claims and aim at exact asymptotic formulas. We establishlocal uniformity for the obtained asymptotic formulas for the subexponential case and globaluniformity for the extended-regularly-varying case. More importantly, in comparison withthe corresponding existing results of Tang (2007) and Hao and Tang (2008) in the caseof independent X and �, our formulas successfully capture the impact of the dependencestructure of (X; �).The asymptotic behavior of the �nite-time and in�nite-time ruin probabilities of the re-

newal risk model of standard structure with a constant force of interest r > 0 has beenextensively investigated in the literature. The reader is referred to Asmussen (1998), Klüp-pelberg and Stadtmüller (1998), Kalashnikov and Konstantinides (2000), Konstantinides etal. (2002), Tang (2005), and Wang (2008). It is worthwhile noting that, if both the forceof interest r > 0 and the premium rate c � 0 are constant and the claim-size distribution issubexponential, then the tail probability of the discounted aggregate claims up to a �nite orin�nite time is asymptotically equivalent to the probability of ruin by the same time. Thisis because the amount of discounted aggregate premiums is always bounded by a �nite con-stant c=r and, thus, it does not a¤ect the asymptotic behavior of a subexponential tail. Dueto this reason, our results in this paper can be straightforwardly translated in the form of

3

�nite-time and in�nite-time ruin probabilities. In comparison with the corresponding resultsof the works cited above, our formulas also explicitly show the impact of the dependencestructure of (X; �) on ruin.The rest of this paper consists of four sections. Section 2 shows two main results after

brie�y introducing necessary preliminaries about the renewal risk model and subexponentialdistributions, Section 3 veri�es the local and global uniformity of relation (1.1) throughcopulas, and Sections 4 and 5 prove the two theorems, respectively.

2 Main results

Throughout this paper, all limit relationships hold as x ! 1 unless stated otherwise. Fortwo positive functions a(�) and b(�) satisfying

l1 = lim infx!1

a(x)

b(x)� lim sup

x!1

a(x)

b(x)= l2;

we write a(x) & b(x) if l1 � 1, write a(x) . b(x) if l2 � 1, write a(x) � b(x) if l1 = l2 = 1,and write a(x) � b(x) if 0 < l1 � l2 < 1. Very often we equip limit relationships withcertain uniformity, which is crucial for our purpose. For instance, for two positive bivariatefunctions a(�; �) and b(�; �), we say that a(x; t) . b(x; t) holds uniformly for t 2 � 6= ? if

lim supx!1

supt2�

a(x; t)

b(x; t)� 1:

Consider the renewal risk model in which (Xk; �k), k = 1; 2; :::, are i.i.d. copies of ageneric pair (X; �) with marginal distributions F and G on [0;1). To avoid triviality, bothF and G are assumed not degenerate at 0. Denote by � k =

Pki=1 �i, k = 1; 2; :::, the claim

arrival times, with � 0 = 0. Then the number of claims by time t is

Nt = #fk = 1; 2; : : : : � k � tg; t � 0;

which forms an ordinary renewal counting process with a �nite mean function

�t = ENt =

1Xk=1

Pr (� k � t) ; t � 0:

In this way, the amount of aggregate claims is a random sum of the form X(t) =PNt

k=1Xk

for t � 0, where and throughout a summation over an empty index set produces a value 0.Assuming a constant force of interest r � 0, the amount of discounted aggregate claims bytime t is expressed as

Dr(t) =

Z t

0�e�rsdX(s) =

1Xk=1

Xke�r�k1(�k�t); t � 0; (2.1)

where the symbol 1E denotes the indicator function of an event E.

4

When studying the tail probability of Dr(t) it is natural to restrict the region of thevariable t to

� = ft : 0 < �t � 1g :

With t = infft : Pr (� � t) > 0g, it is clear that � = [t;1] if Pr (� = t) > 0 while � = (t;1]if Pr (� = t) = 0. For notational convenience, we write �T = [0; T ]\� for every �xed T 2 �.We only consider the case of heavy-tailed claim-size distributions. One of the most

important classes of heavy-tailed distributions is the class S of subexponential distributions.By de�nition, a distribution F on [0;1) is said to be subexponential if F (x) = 1�F (x) > 0for all x � 0 and the relation

limx!1

F n�(x)

F (x)= n

holds for all (or, equivalently, for some) n = 2; 3; : : : ; where F n� denotes the n-fold convolu-tion of F . The class S contains a lot of important distributions such as Pareto, lognormal,and heavy-tailed Weibull distributions. See Embrechts et al. (1997) for a nice review ofsubexponential distributions in the context of insurance and �nance.A useful subclass of S is the class of distributions with extended-regularly-varying (ERV)

tails, characterized by the relations F (x) > 0 for all x � 0 and

y�� � lim infx!1

F (xy)

F (x)� lim sup

x!1

F (xy)

F (x)� y��; y � 1; (2.2)

for some 0 < � � � <1. We signify the regularity property in (2.2) as F 2 ERV(��;��),so that ERV is the union of all ERV(��;��) over the range 0 < � � � <1. In particular,when � = �, the class ERV(��;��) coincides with the famous class R�� of distributionswith regularly-varying tails. Thus, if F 2 R�� for some 0 < � <1 then

limx!1

F (xy)

F (x)= y��; y > 0: (2.3)

Write R as the union of all R�� over the range 0 < � <1.For a distribution F 2 ERV(��;��) for some 0 < � � � < 1, by Proposition 2.2.3 of

Bingham et al. (1989) we know that, for every " > 0 and b > 1, there is some x0 > 0 suchthat the inequalities

1

b

�y���" ^ y��+"

�� F (xy)

F (x)� b

�y���" _ y��+"

�(2.4)

hold whenever x > x0 and xy > x0. In particular, when � = � the inequalities in (2.4)reduce to the well-known Potter�s bounds for the class R. Precisely, if F 2 R�� for some0 < � <1, then for every " > 0 and b > 1, there is some x0 > 0 such that the inequalities

1

b

�y���" ^ y��+"

�� F (xy)

F (x)� b

�y���" _ y��+"

�(2.5)

5

hold whenever x > x0 and xy > x0. In addition, by Theorem 1.5.2 of Bingham et al. (1989),the convergence in relation (2.3) is uniform for y 2 [";1) for every �xed " > 0; that is tosay,

limx!1

supy2[";1)

����F (xy)F (x)� y��

���� = 0: (2.6)

As mentioned in Section 1, a standing assumption on the dependence structure of (X; �)in this paper is the following:

A1: There is some measurable function h(�) : [0;1) 7�! (0;1) such that relation (1.1)holds locally uniformly for t 2 � (that is to say, it holds uniformly for t 2 �T for everyT 2 �).

To achieve global uniformity of the obtained asymptotic formula, we need to strengthenAssumption A1 to the following:

A2: There is some measurable function h(�) : [0;1) 7�! (0;1) such that relation (1.1)holds uniformly for t 2 �.

In addition to Assumption A1 or A2, we also need to assume the following:

B: Either t > 0, or t = 0 and there is some t� 2 � such that inf0�t�t� h(t) > 0.

Note that t� appearing in Assumption B can be chosen to be 0 if Pr(� = 0) > 0, and in thiscase the restriction inf0�t�t� h(t) > 0 is redundant since h(0) > 0 by Assumption A1 or A2.We remark that these assumptions on the dependence structure of (X; �) are close to

minimum for establishing a locally or globally uniform asymptotic formula.The �rst main result of this paper is given below:

Theorem 2.1. Consider the discounted aggregate claims described in relation (2.1) withr � 0. If F 2 S and Assumptions A1 and B hold, then the relation

Pr (Dr(t) > x) �Z t

0�F (xers) d~�s (2.7)

holds locally uniformly for t 2 �, where

~�t =

Z t

0�(1 + �t�u)h(u)G(du): (2.8)

If, in addition to the other conditions of Theorem 2.1, F 2 R�� for some 0 < � < 1,then applying the uniformity of relation (2.3) as explained in (2.6) to relation (2.7), weobtain that the relation

Pr (Dr(t) > x) � F (x)Z t

0�e��rsd~�s (2.9)

holds locally uniformly for t 2 �.

6

Consider the uniformity of relation (2.7) on �T for some T 2 �. Under AssumptionA1, integrating both sides of (1.1) with respect to Pr(� 2 dt) over the range [0; T ] leads to0 < Eh(�)1(��T ) � 1. Similarly as in (1.2), we introduce an independent random variable ��with a proper distribution given by

Pr(�� 2 dt) = h(t)

Eh(�)1(��T )G(dt); t 2 [0; T ]:

Construct a delayed renewal counting process fN�t ; t � 0g with inter-arrival times ��, �k,

k = 2; 3 : : :, and a mean function ��t . It is easy to see that

~�t = ��tEh(�)1(��T ); t 2 �T ;

that is to say, ~�t is proportional to the mean function of a delayed renewal counting processwhose �rst inter-arrival time is a¤ected by the dependence structure of (X; �).Next we establish global uniformity for relation (2.7). For this purpose, we need to

restrict the claim-size distribution to the class ERV. Notice that if r = 0 then Dr(t) divergesto1 almost surely as t!1 and, hence, it is not possible to establish the global uniformityfor relation (2.7). For this reason, we assume r > 0 in the following second main result:

Theorem 2.2. Consider the discounted aggregate claims described in relation (2.1) withr > 0. If F 2 ERV and Assumptions A2 and B hold, then relation (2.7) holds uniformlyfor t 2 �.

Similarly as above, if, in addition to the other conditions of Theorem 2.2, F 2 R�� forsome 0 < � < 1, then, by (2.5) and (2.6), it is easy to verify that relation (2.9) holdsuniformly for t 2 �. In particular, taking t =1 in relation (2.9) yields a more transparentasymptotic formula that

Pr (Dr(1) > x) � F (x)Eh(�)e��r�

1� Ee��r�:

Moreover, under Assumption A2, which implies Eh(�) = 1, ~�t is equal to the meanfunction of a delayed renewal counting process whose �rst inter-arrival time �� follows Pr(�� 2dt) = h(t)G(dt).When X and � are independent, h(t) � 1 and ~�t � �t for all t 2 �. Hence, Theorem 2.1

of Hao and Tang (2008) corresponds to our Theorem 2.1 for the case of independent X and�. The expression of ~�t given in (2.8) for the general case of dependent X and � shows thatour results successfully capture the impact of the dependence structure of (X; �) on the tailbehavior of the discounted aggregate claims.Asimit and Badescu (2010) studied the same problem. Our work extends theirs in the

following three directions: (i) they considered the compound Poisson risk model while weconsider the renewal risk model; (ii) they derived results for F 2 S when r = 0 and forF 2 R when r > 0, both of which are covered and uni�ed by our Theorem 2.1; (iii) theirformulas hold for a �xed time t while ours are equipped with local or global uniformity intime t, which greatly enhances the theoretical and applied interests of the results.

7

Restricted to the compound Poisson risk model with F 2 R�� for some 0 < � < 1,assuming the Poisson intensity � > 0 our formula (2.9) immediately gives

Pr (Dr(t) > x) � Kr;tF (x) ; t > 0; (2.10)

with

Kr;t =

Z t

0

��1 +

�

�r

�e��ru � �

�re��rt

�h(u)�e��udu:

Theorem 3.2 of Asimit and Badescu (2010) also gives relation (2.10) but with a coe¢ cient

K�r;t =

1Xn=1

e��t�nZ� � �Z

n;t

nXi=1

h(si)e��r

Pij=1 sj

nYi=1

dsi;

where n;t = f(s1; : : : ; sn) 2 [0; t]n :Pn

i=1 si � tg. It is not hard to verify that the twocoe¢ cients are actually the same though they look quite di¤erent. Indeed, recall that, forthe Poisson process fNt; t � 0g, the inter-arrival times �1, : : :, �n conditional on (Nt = n)have a joint distribution on n;t given by

Pr (�1 2 ds1; : : : ; �n 2 dsnjNt = n) =n!

tnds1 � � � dsn:

Under the help of this property, we have

K�r;t =

1Xn=1

(�t)n

n!e��t

Z� � �Z

n;t

nXi=1

h(si)e��r

Pij=1 sj

n!

tn

nYi=1

dsi

=1Xn=1

Pr (Nt = n) E

nXi=1

h(�i)e��r� i

�����Nt = n!

=1Xn=1

nXi=1

Eh(�i)e��r� i1(Nt=n)

=1Xi=1

Eh(�i)e��r� i1(� i�t)

=

Z t

0

h(u)e��ruG(du) +1Xi=2

Z t

0

Z t�u

0

h(u)e��r(u+v) Pr (� i�1 2 dv)G(du)

=

Z t

0

h(u)e��ru��e��udu

�+

Z t

0

Z t�u

0

h(u)e��r(u+v) (�dv)��e��udu

�:

Thus, K�r;t = Kr;t.

3 Veri�cation of the assumptions on dependence

This section concerns veri�cation of our assumptions on the dependence structure of (X; �).We carry on this discussion through copulas. The reader is referred to Joe (1997) or Nelsen(2006) for a comprehensive treatment of copulas.

8

For simplicity, assume that H, the joint distribution of (X; �), has continuous marginaldistributions F andG. Then, by Sklar�s theorem, there is a unique copula C(u; v) : [0; 1]2 7�![0; 1], which is the joint distribution of the uniform variates F (X) and G(�), such that

H(x; t) = C (F (x); G(t)) :

The corresponding survival copula is de�ned to be

C(u; v) = u+ v � 1 + C (1� u; 1� v) ; (u; v) 2 [0; 1]2;

which is such that

H(x; t) = Pr (X > x; � > t) = C�F (x); G(t)

�:

Assume that the copula C(u; v) is absolutely continuous, so is the survival copula C(u; v).Denote by c(u; v) the density of the survival copula. Then the function h(�) de�ned in (1.1),if it exists, is equal to

h (t) = limu!0+

@@vC(u; v)

u

�����v=G(t)

= c�0+; G(t)

�; t > 0: (3.1)

In the rest of this section, the variables t and v are always connected through the identityv = G(t), as indicated in (3.1). In terms of the survival copula C(u; v), the local uniformityof relation (1.1), as required by Assumption A1, can be restated as

limu!0+

supv2[�;1]

����� @@v C(u; v)uh(t)� 1����� = lim

u!0+supv2[�;1]

���� 1uR u0c (s; v) ds

c (0+; v)� 1���� = 0; � 2 (0; 1); (3.2)

and the global uniformity of relation (1.1), as required by Assumption A2, can be restatedas

limu!0+

supv2(0;1]

����� @@v C(u; v)uh(t)� 1����� = lim

u!0+supv2(0;1]

���� 1uR u0c (s; v) ds

c (0+; v)� 1���� = 0: (3.3)

So far, the function h(�) and the local/global uniformity of relation (1.1) have been expressedthrough the survival copula and its density.Recall that an Archimedean copula is of the form

C(u; v) = '[�1] ('(u) + '(v)) ; (u; v) 2 [0; 1]2;

where '(�) : [0; 1] 7�! [0;1], called the generator of C(u; v), is a strictly decreasing andconvex function with 0 < ' (0) � 1 and ' (1) = 0, while '[�1](�) is the pseudo-inverse of'(�), equal to '�1(t) when 0 � t � '(0) and equal to 0 otherwise. This copula has a simplestructure, as its de�nition shows, and it enjoys a lot of nice properties. If the generator '(�)is twice di¤erentiable, then the copula density has a transparent form; see relation (4.3.6) ofNelsen (2006). In this case, recalling (3.1), it follows that

h(t) = c (0+; v) = �'0(1�)'00 (1� v)('0(1� v))2

:

9

Moreover, it should not be hard to construct some general conditions on the generator '(�)to guarantee relations (3.2) and (3.3).Next, we reexamine the three examples given in Asimit and Badescu (2010).

Example 3.1. The Ali-Mikhail-Haq copula is of the form

C(u; v) =uv

1� (1� u)(1� v) ; 2 [�1; 1):

Direct calculation shows

@C(u; v)

@v=u+ u(1� 2v)� u2 (1� v2)

(1� uv)2: (3.4)

Then, by relation (3.1),h(t) = 1 + (1� 2v) :

It follows that ����� @C(u;v)@v

uh(t)� 1����� = jv (2� uv) (1 + (1� 2v))� 1 + v2j

(1� uv)2 (1 + (1� 2v))j ju:

Hence, relation (3.2) holds when 2 [�1; 1) and relation (3.3) holds when 2 (�1; 1). �

Example 3.2. The Farlie-Gumbel-Morgenstern copula is of the form

C(u; v) = uv + uv(1� u)(1� v); 2 [�1; 1]:

Going along the same lines of Example 3.1, we have, respectively,

@C(u; v)

@v= u+ u (1� u) (1� 2v);

h(t) = 1 + (1� 2v) ;

and ����� @C(u;v)@v

uh(t)� 1����� = j1� 2vj

1 + (1� 2v) j ju:

Hence, relation (3.2) holds when 2 [�1; 1) and relation (3.3) holds when 2 (�1; 1). �

Example 3.3. The Frank copula is of the form

C(u; v) = �1 ln

�1 +

(e� u � 1) (e� v � 1)e� � 1

�; 6= 0:

Direct but a bit tedious calculation gives

@C(u; v)

@v=

e (1� e u)e (1� e ) + (e u � e ) (e v � e ) :

10

It follows that

h(t) = e (1�v)

e � 1and ����� @C(u;v)@v

uh(t)� 1����� =

�� u �e (1�v) (e u � 1) + (e � e u)�� (e u � 1) (e � 1)�� u (e (1�v) (e u � 1) + (e � e u)) :

Hence, relations (3.2) and (3.3) hold for all 6= 0. �

4 Proof of Theorem 2.1

4.1 Lemmas

It is well known that every subexponential distribution F is long tailed, denoted as F 2 L,in the sense that the relation

limx!1

F (x� y)F (x)

= 1 (4.1)

holds for all (or, equivalently, for some) y 6= 0; see Lemma 2 of Chistyakov (1964) or Lemma1.3.5(a) of Embrechts et al. (1997).We �rst show an elementary result regarding long-tailed distributions:

Lemma 4.1. F 2 L if and only if there is a function l(�) : (0;1) 7�! (0;1) satisfying(i) l(x) < x=2 for all x > 0,(ii) l(x)!1,(iii) l(�) is slowly varying at in�nity,

such that, for every K > 0,F (x�Kl(x)) � F (x):

Proof. The �if�assertion is trivial, so we only prove the �only if�assertion. Let F 2 L. Itis easy to see that there is a positive function l1(�) satisfying l1(x) ! 1, l1(x) < x2=4 forall x > 0, and F (x� l1(x)) � F (x). Furthermore, by Lemma 3.2 of Tang (2008), there is aslowly varying function l(�) : (0;1) 7�! (0;1) satisfying l(x) ! 1 and l(x) � l1(x)1=2 forall x > 0. This function l(�) ful�lls all the requirements in Lemma 4.1.

Lemma 4.2 below forms the main ingredient of the proof of Theorem 2.1. It deals withthe tail probability of the sum of n random variables equipped with a certain dependencestructure. A similar problem was considered in Proposition 2.1 of Foss and Richards (2010).However, a close look tells that their Proposition 2.1 and our Lemma 4.2 below are essentiallydi¤erent.Let X1, . . . , Xn be i.i.d. random variables with common distribution F 2 S. Recall

Proposition 5.1 of Tang and Tsitsiashvili (2003), which shows that, for arbitrarily �xed0 < a � b <1, the relation

Pr

nXk=1

wkXk > x

!�

nXk=1

Pr (wkXk > x) (4.2)

11

holds uniformly for (w1; : : : ; wn) 2 [a; b]n. Hence, for the case of independent X and �,Lemma 4.2 below immediately follows by conditioning on (�1; : : : ; �n). However, for thegeneral case of dependent X and �, this lemma is a nontrivial consequence of Proposition5.1 of Tang and Tsitsiashvili (2003).Hereafter, for notational convenience, for every t 2 � and n = 1; 2; : : :, we write tn =Pni=1 si and n;t = f(s1; : : : ; sn) 2 [0; t]n : tn � tg.

Lemma 4.2. Recall the renewal risk model introduced in Section 2 with r � 0. If F 2 Sand Assumption A1 holds, then for every n = 1; 2; : : :, it holds locally uniformly for t 2 �that

Pr

nXk=1

Xke�r�k > x;Nt = n

!= (1 + o(1))

nXk=1

Pr�Xke

�r�k > x;Nt = n�: (4.3)

Proof. Note that the event (Nt = n) in relation (4.3) could have a probability 0. In theproof below, we still use the notation a(x; t) � b(x; t) even though the two functions a(x; t)and b(x; t) could be simultaneously equal to 0. For such an occasion the notation exactlymeans a(x; t) = (1 + o(1))b(x; t). Thus, doing so will cause no confusion.We need to prove that relation (4.3) holds uniformly for t 2 �T for arbitrarily �xed

T 2 �. Let us proceed the proof by induction. Trivially, the assertion holds for n = 1. Nowwe assume by induction that the assertion holds for some positive integer n = m� 1 and weprove it for n = m; that is to say, we aim at the relation

Pr

mXk=1

Xke�r�k > x;Nt = m

!�

mXk=1

Pr�Xke

�r�k > x;Nt = m�

(4.4)

with the required uniformity for t 2 �T .Recall the function l(�) speci�ed in Lemma 4.1. According to the value of the sumPm�1k=1 Xke

�r�k belonging to (0; l(x)], (x� l(x);1), and (l(x); x� l(x)], we split the proba-bility on the left-hand side of (4.4) into three parts as

Pr

mXk=1

Xke�r�k > x;Nt = m

!= I1(x;m; t) + I2(x;m; t) + I3(x;m; t):

For I1(x;m; t), it holds uniformly for t 2 �T that

I1(x;m; t) � Pr�Xme

�r�m > x� l(x); Nt = m�

=

Z� � �Z

m;t

Pr�Xme

�rtm > x� l(x)�� �m = sm�G (t� tm) mY

i=1

G(dsi)

�Z� � �Z

m;t

Pr�Xme

�rtm > x�h(sm)G (t� tm)

mYi=1

G(dsi)

�Z� � �Z

m;t

Pr�Xme

�rtm > x�� �m = sm�G (t� tm) mY

i=1

G(dsi)

= Pr�Xme

�r�m > x;Nt = m�; (4.5)

12

where at the third and fourth steps we used Assumption A1 and Lemma 4.1. As it showsfrom the second step to the last step, the derivation of (4.5) is mainly to eliminate the slowlyvarying function l(x). For I2(x;m; t), by the induction assumption and the same idea as inderiving (4.5), we have, uniformly for t 2 �T ,

I2(x;m; t) � Pr m�1Xk=1

Xke�r�k > x� l(x); Nt = m

!

=

Z t

0�Pr

m�1Xk=1

Xke�r�k > x� l(x); Nt�sm = m� 1

!G(dsm)

�m�1Xk=1

Z t

0�Pr�Xke

�r�k > x� l(x); Nt�sm = m� 1�G(dsm)

�m�1Xk=1

Z t

0�Pr�Xke

�r�k > x;Nt�sm = m� 1�G(dsm)

=m�1Xk=1

Pr�Xke

�r�k > x;Nt = m�: (4.6)

Now we focus on I3(x;m; t). It holds uniformly for t 2 �T that

I3(x;m; t) � Pr m�1Xk=1

Xke�r�k +Xme

�r�m > x;m�1Xk=1

Xke�r�k > l(x); Xme

�r�m > l(x); Nt = m

!

=

Z t

0�

Z 1

l(x)

Pr

m�1Xk=1

Xke�r�k > (x� y) _ l(x); Nt�sm = m� 1

!Pr�Xme

�rsm 2 dy�� �m = sm�G(dsm)

�m�1Xk=1

Z t

0�

Z 1

l(x)

Pr�Xke

�r�k > (x� y) _ l(x); Nt�sm = m� 1�

Pr�Xme

�rsm 2 dy�� �m = sm�G(dsm)

�m�1Xk=1

Z� � �Z

m;t

Pr�Xke

�rtk +Xme�rsm > x;Xke

�rtk > l(x); Xme�rsm > l(x)

�G(t� tm)h(sk)h(sm)

mYi=1

G(dsi);

where at the third step we used induction assumption and at the last step we applied Assump-tion A1 twice. For every k = 1; : : : ;m�1, we notice that, uniformly for (s1; : : : ; sm) 2 m;t,

Pr�Xke

�rtk +Xme�rsm > x;Xke

�rtk > l(x); Xme�rsm > l(x)

�� Pr

�Xke

�rtk +Xme�rsm > x

��Pr

�Xke

�rtk > x;Xme�rsm � l(x)

�� Pr

�Xme

�rsm > x;Xke�rtk � l(x)

�= o(1)

�Pr�Xke

�rtk > x�+ Pr

�Xme

�rsm > x��;

13

where at the last step we applied Proposition 5.1 of Tang and Tsitsiashvili (2003) as sum-marized in (4.2) above. Plugging these estimates into I3(x;m; t), we have, uniformly fort 2 �T ,

I3(x;m; t) = o(1)

m�1Xk=1

Z� � �Z

m;t

Pr�Xke

�rtk > x�G (t� tm)h(sk)h(sm)

mYi=1

G(dsi)

+o(1)

m�1Xk=1

Z� � �Z

m;t

Pr�Xme

�rsm > x�G (t� tm)h(sk)h(sm)

mYi=1

G(dsi):

Since the distribution F has an ultimate tail, for an arbitrary function a(x) = o(1), we canalways �nd some positive function l�(x), which diverges to 1 but slowly enough, such that

a(x) = o(1)F (l�(x)) :

Using this idea and Assumption A1, we have, uniformly for t 2 �T ,

I3(x;m; t) = o(1)m�1Xk=1

Z� � �Z

m;t

Pr�Xke

�rtk > x;Xm > l�(x)

�G (t� tm)h(sk)h(sm)

mYi=1

G(dsi)

+o(1)m�1Xk=1

Z� � �Z

m;t

Pr�Xme

�rsm > x;Xk > l�(x)

�G (t� tm)h(sk)h(sm)

mYi=1

G(dsi)

= o(1)m�1Xk=1

Pr�Xke

�r�k > x;Xm > l�(x); Nt = m

�+o(1)

m�1Xk=1

Pr�Xme

�r�m > x;Xk > l�(x); Nt = m

�= o(1)

mXk=1

Pr�Xke

�r�k > x;Nt = m�: (4.7)

A combination of (4.5), (4.6), and (4.7) gives the upper-bound version of (4.4).The corresponding lower-bound version of (4.4) can be easily established. Actually,

Pr

mXk=1

Xke�r�k > x;Nt = m

!

� Pr

m�1Xk=1

Xke�r�k > x

![�Xme

�r�m > x�; Nt = m

!

= Pr

m�1Xk=1

Xke�r�k > x;Nt = m

!+ Pr

�Xme

�r�m > x;Nt = m�

�Pr m�1Xk=1

Xke�r�k > x;Xme

�r�m > x;Nt = m

!= J1(x;m; t) + Pr

�Xme

�r�m > x;Nt = m�� J2(x;m; t): (4.8)

14

As in dealing with I2(x;m; t), by the induction assumption, we have, uniformly for t 2 �T ,

J1(x;m; t) �m�1Xk=1

Pr�Xke

�r�k > x;Nt = m�: (4.9)

Furthermore, as in dealing with I3(x;m; t), it holds uniformly for t 2 �T that

J2(x;m; t) � Pr m�1Xk=1

Xke�r�k+Xme

�r�m > x;m�1Xk=1

Xke�r�k > l(x); Xme

�r�m > l(x); Nt = m

!

= o(1)mXk=1

Pr�Xke

�r�k > x;Nt = m�: (4.10)

Plugging (4.9) and (4.10) into (4.8) gives the lower-bound version of (4.4). This ends theproof of Lemma 4.2.

Going along the same lines of the proof of Lemma 4.2 with some obvious modi�cations,we can obtain the following result:

Lemma 4.3. Under the same conditions of Lemma 4.2, for every n = 1; 2; : : :, it holdslocally uniformly for t 2 � that

Pr

nXk=1

Xke�r�k > x; �n � t

!= (1 + o(1))nPr

�X1e

�r�1 > x; �n � t�: (4.11)

For a distribution F 2 S, the well-known Kesten�s inequality states that, for every " > 0,there is some constant K = K" > 0 such that the inequality

F n�(x) � K(1 + ")nF (x)

holds for all n = 1; 2; : : : and x � 0. For its proof see Athreya and Ney (1972) or Lemma1.3.5(c) of Embrechts et al. (1997). In the following lemma, we establish an inequality ofKesten�s type for the probability on the left-hand side of (4.11):

Lemma 4.4. Recall the renewal risk model introduced in Section 2 with r � 0. If F 2 S,t = 0, and Assumptions A1 and B hold, then for every " > 0 and T 2 �, there is someconstant K = Kr;";T > 0 such that the inequality

Pr

nXk=1

Xke�r�k > x; �n � t

!� K(1 + ")n Pr

�X1e

�r�1 > x; �n � t�

holds for all n = 1; 2; : : :, x � 0, and t 2 �T .

Proof. For every " > 0, by Lemma 4.3, there is some constant x0 > 0 such that, for allx > x0 and t 2 �T ,

Pr�X1e

�r�1 +X2e�r�2 > x;X2e

�r�2 � x; � 2 � t�

= Pr�X1e

�r�1 +X2e�r�2 > x; � 2 � t

�� Pr

�X2e

�r�2 > x; � 2 � t�

� (1 + ") Pr�X1e

�r�1 > x; � 2 � t�: (4.12)

15

By Assumption A1, the constant x0 above can be chosen so large that, for all t 2 �T ,

Pr�X1e

�r�1 > x0j�1 = t�� 1

2F�x0e

rT�h(t): (4.13)

Write

an = supx�0t2�T

Pr�Pn

k=1Xke�r�k > x; �n � t

�Pr (X1e�r�1 > x; �n � t)

:

We start to evaluate an+1. It is clear that

Pr

n+1Xk=1

Xke�r�k > x; �n+1 � t

!

= Pr

n+1Xk=1

Xke�r�k > x;Xn+1e

�r�n+1 � x; �n+1 � t!+ Pr

�Xn+1e

�r�n+1 > x; �n+1 � t�:

Conditioning on (Xn+1; �n+1) and noticing the de�nition of an, we have

Pr

n+1Xk=1

Xke�r�k > x;Xn+1e

�r�n+1 � x; �n+1 � t!

� an Pr�X1e

�r�1 +Xn+1e�r�n+1 > x;Xn+1e

�r�n+1 � x; �n+1 � t�:

Using (4.12) we know that, for all x > x0 and t 2 �T ,

Pr�X1e

�r�1 +Xn+1e�r�n+1 > x;Xn+1e

�r�n+1 � x; �n+1 � t�

=

Z� � �Z

n�1;t

Pr�X1e

�r�1 +X2e�r�2 > x;X2e

�r�2 � x; � 2 � t� tn�1� n�1Yi=1

G(dsi)

� (1 + ") Pr�X1e

�r�1 > x; �n+1 � t�:

Hence,

supx>x0t2�T

Pr�Pn+1

k=1 Xke�r�k > x; �n+1 � t

�Pr (X1e�r�1 > x; �n+1 � t)

� (1 + ")an + 1: (4.14)

When x � x0, by inequality (4.13), it holds for all t 2 �T that

Pr�Pn+1

k=1 Xke�r�k > x; �n+1 � t

�Pr (X1e�r�1 > x; �n+1 � t)

� Pr (�n+1 � t)Pr (X1e�r�1 > x0; �n+1 � t)

��1

2F�x0e

rT���1 R t

0�Gn�(t� s)G(ds)R t

0�Gn�(t� s)h(s)G(ds)

:

By Assumption B, there is some constant 0 � t� 2 � such that h(s) is away from 0 fors 2 [0; t�]. We haveR t

0�Gn�(t� s)G(ds)R t

0�Gn�(t� s)h(s)G(ds)

�R t^t�0� Gn�(t� s)G(ds) +

R tt^t� G

n�(t� s)G(ds)1(t>t�)R t^t�0� Gn�(t� s)h(s)G(ds)

��inf

s2[0;t�]h(s)

��1+

G(t�)R t�0� h(s)G(ds)

:

16

Therefore, there is some 0 < L <1 such that

supx�x0t2�T

Pr�Pn+1

k=1 Xke�r�k > x; �n+1 � t

�Pr (X1e�r�1 > x; �n+1 � t)

� L: (4.15)

It follows from (4.14) and (4.15) that

an+1 � (1 + ")an + 1 + L:

This recursive inequality with initial value a1 = 1 completes the proof of Lemma 4.4.

4.2 Proof of Theorem 2.1

We follow the proof of Theorem 2.1 of Hao and Tang (2008) but we need to overcome sometechnical di¢ culties due to the dependence structure of (X; �).Let us prove that relation (2.7) holds uniformly for t 2 �T for arbitrarily �xed T 2 �.

Choose some large positive integer N and write

Pr (Dr(t) > x) =

1Xn=N+1

+NXn=1

!Pr

nXk=1

Xke�r�k > x;Nt = n

!= I1(x; t)+I2(x; t): (4.16)

First, we look at I1(x; t). Note that if t > 0 then I1(x; t) vanishes for some large N . Thus,we can assume t = 0. Applying Lemma 4.4, for every " > 0, there is some constant K > 0

such that, for all t 2 �T ,

I1(x; t) �1X

n=N+1

Pr

nXk=1

Xke�r�k > x; �n � t

!

� K1X

n=N+1

(1 + ")n Pr�X1e

�r�1 > x; �n � t�:

By Assumption A1, it follows that, uniformly for t 2 �T ,

I1(x; t) . K1X

n=N+1

(1 + ")nZ t

0�F (xers1)h(s1) Pr (Nt�s1 � n� 1)G(ds1)

� KZ t

0�F (xers)h(s)G(ds)

1Xn=N+1

(1 + ")n Pr (NT � n� 1) :

It is well known that the moment generating function of NT is analytic in a neighborhoodof 0; see, e.g. Stein (1946). Thus, we may choose some " > 0 su¢ ciently small such thatthe series at the last step above converges. Therefore, for every 0 < � < 1, we can �nd somelarge positive integer N such that, uniformly for t 2 �T ,

I1(x; t) . �Z t

0�F (xers) d~�s: (4.17)

17

Next, we turn to I2(x; t). By Lemma 4.2, it holds uniformly for t 2 �T that

I2(x; t) � 1Xn=1

�1X

n=N+1

!nXk=1

Pr�Xke

�r�k > x;Nt = n�= I21(x; t)� I22(x; t): (4.18)

For I21(x; t), by interchanging the order of the sums then conditioning on � k�1 and �k, weobtain that, uniformly for t 2 �T ,

I21(x; t) =

1Xk=1

Pr�Xke

�r�k > x; � k � t�

�1Xk=1

Z t

0�

Z t�u

0�F�xer(u+v)

�Pr (� k�1 2 dv)h(u)G(du)

=

Z t

0�

�F (xeru) +

Z t�u

0�F�xer(u+v)

�d�v

�h(u)G(du)

=

Z t

0�F (xers)h(s)G(ds) +

Z t

0�F (xers) d

�Z s

0��s�uh(u)G(du)

�=

Z t

0�F (xers) d~�s; (4.19)

where at the fourth step we used integration by parts with possible jumps; see, e.g. formula(1.20) of Klebaner (2005). For I22(x; t), it holds uniformly for t 2 �T that

I22(x; t) =1X

n=N+1

nXk=1

Pr�Xke

�r�k > x;Nt = n�

.1X

n=N+1

nXk=1

Z t

0�F (xersk) Pr (Nt�sk = n� 1)h (sk)G(dsk)

=1X

n=N+1

n

Z t

0�F (xers) Pr (Nt�s = n� 1)h (s)G(ds)

� E (1 +NT )1(NT�N)Z t

0�F (xers)h (s)G(ds):

Hence, we can �nd some large positive integer N such that, uniformly for t 2 �T ,

I22(x; t) . �Z t

0�F (xers) d~�s: (4.20)

Plugging (4.19) and (4.20) into (4.18) yields that, uniformly for t 2 �T ,

(1� �)Z t

0�F (xers) d~�s . I2(x; t) .

Z t

0�F (xers) d~�s: (4.21)

Plugging (4.17) and (4.21) into (4.16) and noticing the arbitrariness of �, we complete theproof of Theorem 2.1. �

18

5 Proof of Theorem 2.2

5.1 Lemmas

The following lemma describes closure of the class ERV under the product of two dependentrandom variables:

Lemma 5.1. Recall the renewal risk model introduced in Section 2 with r � 0. If F 2ERV(��;��) for some 0 < � � � < 1 and Assumption A2 holds, then for every k =1; 2; : : :, the distribution of Xke

�r�k still belongs to ERV(��;��) and

Pr�Xke

�r�k > x�� F (x): (5.1)

Proof. For every k = 1; 2; : : :, conditioning on �k, � k�1 and recalling Assumption A2, wehave

Pr�Xke

�r�k > x��Z 1

0�

Z 1

0�F�xer(u+v)

�h(u)G(du) Pr (� k�1 2 dv) : (5.2)

It follows from (5.2) and (2.2) that, for every y � 1,

lim supx!1

Pr (Xke�r�k > xy)

Pr (Xke�r�k > x)� lim sup

x!1sups�0

F (xyers)

F (xers)� y��

and

lim infx!1

Pr (Xke�r�k > xy)

Pr (Xke�r�k > x)� lim inf

x!1infs�0

F (xyers)

F (xers)� y��:

Therefore, the distribution of Xke�r�k belongs to ERV(��;��). Moreover, applying (2.4)

to (5.2), we obtain relation (5.1).

The following lemma is interesting in its own right and it will be the main ingredient ofthe proof of Theorem 2.2:

Lemma 5.2. Recall the renewal risk model introduced in Section 2 with r > 0. If F 2ERV(��;��) for some 0 < � � � <1 and Assumption A2 holds, then it holds uniformlyfor n = 1; 2; : : : that

Pr

nXk=1

Xke�r�k > x

!�

nXk=1

Pr�Xke

�r�k > x�: (5.3)

Proof. By Lemma 5.1, the distributions of Xke�r�k , k = 1; 2; : : :, belong to ERV(��;��)

and, for 1 � i 6= j � n,

Pr�Xie

�r� i > x;Xje�r�j > x

���F (x)

�2= o

�Pr�Xie

�r� i > x�+ Pr

�Xje

�r�j > x��:

Hence, it follows from Theorem 3.1 of Chen and Yuen (2009) that relation (5.3) holds forevery �xed n = 1; 2; : : :.Now, we turn to the required uniformity of relation (5.3). Trivially, it holds for every

k = n+1; n+2; : : : that Xke�r�k � Xke

�r�k�1, where Xk and e�r�k�1 on the right-hand side

19

are independent. Hence, going along the same lines of the proof of Theorem 3.1 of Tangand Tsitsiashvili (2004) with some obvious modi�cations (see also Section 4 of Chen and Ng2007), we obtain

limn!1

lim supx!1

1

F (x)Pr

1Xk=n+1

Xke�r�k > x

!= lim

n!1lim supx!1

1Xk=n+1

Pr (Xke�r�k > x)

F (x)= 0:

This means that, for every � > 0, there is some large positive integer n0 such that

Pr

1Xk=n0+1

Xke�r�k > x

!+

1Xk=n0+1

Pr�Xke

�r�k > x�. �F (x): (5.4)

By relation (5.3) with n = n0, the �rst assertion of Lemma 5.1, and relation (5.4), it holdsfor arbitrarily �xed 0 < " < 1 and uniformly for n > n0 that

Pr

nXk=1

Xke�r�k > x

!� Pr

n0Xk=1

Xke�r�k > (1� ")x

!+ Pr

1Xk=n0+1

Xke�r�k > "x

!

. (1� ")��n0Xk=1

Pr�Xke

�r�k > x�+ �"��F (x)

� (1� ")��nXk=1

Pr�Xke

�r�k > x�+ �"��F (x): (5.5)

Symmetrically, it holds uniformly for n > n0 that

Pr

nXk=1

Xke�r�k > x

!� Pr

n0Xk=1

Xke�r�k > x

!

�

nXk=1

�nX

k=n0+1

!Pr�Xke

�r�k > x�

&nXk=1

Pr�Xke

�r�k > x�� �F (x): (5.6)

By relation (5.1) and the arbitrariness of " and �, we conclude from (5.5) and (5.6) thatrelation (5.3) holds uniformly for n > n0. The uniformity of relation (5.3) for 1 � n � n0is obvious since it holds for every �xed n = 1; 2; : : :. This completes the proof of Lemma5.2.

5.2 Proof of Theorem 2.2

Following the proof of Lemma 4.2 of Hao and Tang (2008), we have

limt!1

lim supx!1

R1tF (xers)d~�sR t

0� F (xers)d~�s

= 0:

20

Thus, for every � > 0, there is some T0 2 � such thatZ 1

T0

F (xers) d~�s . �Z T0

0�F (xers) d~�s: (5.7)

On the one hand, by Theorem 2.1 and relation (5.7), it holds uniformly for t 2 (T0;1] that

Pr (Dr(t) > x) � Pr (Dr (T0) > x)

��Z t

0��Z t

T0

�F (xers) d~�s

& (1� �)Z t

0�F (xers) d~�s: (5.8)

On the other hand, by Lemma 5.2 and Assumption A2,

Pr (Dr(1) > x) �1Xk=1

Pr�Xke

�r�k > x�

= Pr�X1e

�r�1 > x�+

Z 1

0�Pr�Xke

�r�k > xerv�d�v

�Z 1

0�F (xers)h (s)G(ds) +

Z 1

0�

Z 1

0�F�xer(u+v)

�h(u)G(du)d�v

=

Z 1

0�F (xers) d~�s: (5.9)

Hence, by (5.9) and (5.7), it holds uniformly for t 2 (T0;1] that

Pr (Dr(t) > x) � Pr (Dr(1) > x)

��Z t

0�+

Z 1

t

�F (xers) d~�s

. (1 + �)Z t

0�F (xers) d~�s: (5.10)

By the arbitrariness of � in (5.8) and (5.10), we prove the uniformity of (2.7) for t 2 (T0;1].Recall that Theorem 2.1 already shows the local uniformity of (2.7). This ends the proof ofTheorem 2.2. �

Acknowledgment. The authors are most grateful to the two referees for their very thoroughreading of the paper and valuable suggestions. The research of Jinzhu Li and Rong Wu wassupported by the National Basic Research Program of China (973 Program, Project No:2007CB814905).

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