Uniform asymptotics for the finite-time ruin probability with upper tail asymptotically independent claims and constant force of interest

Statistics and Probability Letters 83 (2013) 1527–1538

Contents lists available at SciVerse ScienceDirect

Statistics and Probability Letters journal homepage: www.elsevier.com/locate/stapro

Uniform asymptotics for the finite-time ruin probability with upper tail asymptotically independent claims and constant force of interest Qingwu Gao a,∗ , Xijun Liu b a

School of Mathematics and Statistics, Nanjing Audit University, Nanjing 211815, China

b

Foundation Department, The First Aeronautical College of Air Force, Xinyang 464000, China

article

info

Article history: Received 10 November 2012 Received in revised form 22 February 2013 Accepted 23 February 2013 Available online 28 February 2013 MSC: 62P05 62E20 60F05

abstract This paper investigates the finite-time ruin probability in a risk model with constant force of interest, upper tail asymptotically independent claims, and a general claim arrival process. We obtain a uniformly asymptotic formula for times in a finite interval. In particular, with a certain dependence among the inter-arrival times, the formula holds uniformly for all times. © 2013 Elsevier B.V. All rights reserved.

Keywords: Uniform asymptotics Finite-time ruin probability Upper tail asymptotic independence Counting process Widely lower orthant dependence

1. Risk model In the paper, we consider the finite-time ruin probability in a nonstandard risk model with the following assumptions. Assumption 1. The claim sizes {Xi , i ≥ 1} form a sequence of nonnegative, but not necessarily independent, random variables (r.v.s) with distributions Fi , i ≥ 1, respectively. Assumption 2. The claim arrival process {N (t ), t ≥ 0} is a general counting process, independent of {Xi , i ≥ 1} and satisfying EN (t ) < ∞ for all 0 < t < ∞. Assumption 3. The total amount of premiums accumulated before time t ≥ 0, denoted by C (t ), is a nonnegative and nondecreasing stochastic process with C (0) = 0 and C (t ) < ∞ almost surely (a.s.) for every 0 < t < ∞. Denote the arrival times of successive claims by τi , i ≥ 1. Then θ1 = τ1 , θi = τi − τi−1 , i ≥ 2, are the claim inter-arrival times. Define Λ = {t : EN (t ) > 0} = {t : P (τ1 ≤ t ) > 0}, and write t = inf{t : EN (t ) > 0} = inf{t : P (τ1 ≤ t ) > 0}.

∗

Corresponding author. Tel.: +86 025 58318699. E-mail addresses: [email protected], [email protected] (Q. Gao).

0167-7152/$ – see front matter © 2013 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.spl.2013.02.018

1528

Q. Gao, X. Liu / Statistics and Probability Letters 83 (2013) 1527–1538

It is clear that

Λ=

 [t , ∞], (t , ∞],

if P (τ1 = t ) > 0, if P (τ1 = t ) = 0.

Let r ≥ 0 be a constant force of interest and x ≥ 0 be an insurer’s initial reserve. Hence, the total reserve up to time t ≥ 0 of the insurance company, denoted by Ur (t ), satisfies Ur (t ) = xert +

t



er (t −s) C (ds) −

0

∞ 

Xi er (t −τi ) 1{τi ≤t } ,

(1.1)

i=1

and the discounted aggregate claims up to time t ≥ 0 are expressed as Dr (t ) =

∞ 

Xi e−r τi 1{τi ≤t } ,

(1.2)

i =1

where 1E is the indicator function of an event E. Obviously, we attain from Assumption 3 that for any fixed 0 < t < ∞, 0 ≤ C (t ) =

t



e−rs C (ds) < ∞

a.s.,

(1.3)

0

where  C (t ) denotes the discounted value of premiums accumulated before time t > 0. As usual, we define the ruin probability within a finite time t > 0 as

ψr (x, t ) = P (Ur (s) < 0 for some 0 ≤ s ≤ t ).

(1.4)

2. Introduction and main results Henceforth, all limit relationships are taken as x → ∞ unless mentioned otherwise. For two positive functions a(·) and b(·) such that C − = lim inf a(x)/b(x) ≤ lim sup a(x)/b(x) = C + , we write a(x) & b(x) if C − ≥ 1, write a(x) . b(x) if C + ≤ 1, write a(x) ∼ b(x) if both, write a(x) = o(1)b(x) if C + = 0, and write a(x) ≍ b(x) if 0 < C − ≤ C + < ∞. For two positive bivariate functions a(·, ·) and b(·, ·), we say that the relation a(x, t ) ∼ b(x, t ) holds uniformly for all t ∈ ∆ ̸= ∅ if limx→∞ supt ∈∆ |a(x, t )/b(x, t ) − 1| = 0. For a distribution F and any y > 0, we define JF+ = − lim log F ∗ (y)/ log y and y→∞

∗

JF− = − lim log F (y)/ log y, y→∞

∗

where F ∗ (y) = lim infx→∞ F (xy)/F (x) and F (y) = lim supx→∞ F (xy)/F (x). We notice that, in the insurance industry practitioners usually choose heavy-tailed r.v.s to model large claims. A natural class of large-claim distributions is the subexponential class. We say that a claim size or its distribution F is subexponential, denoted by F ∈ S , if F (x) = 1 − F (x) > 0 for all x > 0 and F ∗2 (x) ∼ 2F (x), where F ∗2 denotes the 2-fold convolution of F . From Lemma 2.5.1 of Rolski et al. (1999) or Lemma 1.3.5 of Embrechts et al. (1997), we know that if F ∈ S then F is long-tailed, denoted by F ∈ L and characterized by F (x + y) ∼ F (x) for all y ̸= 0.

(2.1)

Another important class is the dominated variation class D , we say that a claim size or its distribution F belongs to the class ∗ D , denoted by F ∈ D , if F (y) < ∞ for all y > 0. A slightly smaller subclass of L ∩ D is the consistent variation class C , we ∗ say that a distribution F belongs to the class C , denoted by F ∈ C , if limy↘1 F ∗ (y) = 1, or equivalently, limy↗1 F (y) = 1. In conclusion,

C ⊂ L ∩ D ⊂ S ⊂ L. For more details of heavy-tailed distributions in the insurance and finance context or literature, the readers are referred to Bingham et al. (1987) and Embrechts et al. (1997). It is well-known that in the standard renewal risk model with constant force of interest, i.i.d. heavy-tailed claim sizes and i.i.d. inter-arrival times, many researchers have devoted themselves to some ruin-related problems, see Tang (2005, 2007), Hao and Tang (2008), and references therein. But in most practical situations, the independence assumption on claim sizes and/or inter-arrival times is unrealistic and then it limits the usefulness of the obtained results to some extent. Recently, more and more attention has been paid to some nonstandard renewal risk models with dependent claim sizes and/or dependent inter-arrival times, for example, see Chen and Ng (2007), Li et al. (2009), Yang and Wang (2010), Liu et al. (2012), Gao et al. (in press), Wang et al. (2013) and others. Besides, a new trend in this study is to allow a certain dependence structure between claim sizes and their arrival process, see Asimit and Badescu (2010), Li et al. (2010) and Chen and Yuen (2012). Therein, Wang et al. (2013) introduced a dependence structure as follows.

Q. Gao, X. Liu / Statistics and Probability Letters 83 (2013) 1527–1538

1529

Definition 2.1. Say that r.v.s {Xi , i ≥ 1} are widely upper orthant dependent (WUOD), if there exists a sequence of finite positive real numbers {gU (n), n ≥ 1} such that for each n ≥ 1 and all xi ∈ (−∞, ∞), 1 ≤ i ≤ n,

 P

n  

Xi > xi



 ≤ gU (n)

n 

P (Xi > xi ).

i=1

i=1

If the above inequality is changed to

 P

n  

X i ≤ xi



 ≤ gL (n)

n 

P (Xi ≤ xi ),

i =1

i=1

where {gL (n), n ≥ 1} is another sequence of finite positive real numbers, then we say that {Xi , i ≥ 1} are widely lower orthant dependent (WLOD). We mention that if {Xi , i ≥ 1} are WLOD, then {−Xi , i ≥ 1} are WUOD, and for each n ≥ 1 and any s > 0, it holds that

 E exp −s

n 

 Xi

≤ g L ( n)

i =1

n 

E exp{−sXi }.

(2.2)

i=1

Additionally, Geluk and Tang (2009) and Liu et al. (2012) also proposed a more general dependence structure, whose definition is given below. Definition 2.2. Say that r.v.s {Xi , i ≥ 1} are upper tail asymptotically independent (UTAI), if P (Xi > x) > 0 for all x ∈ (−∞, ∞), i ≥ 1 and lim

xi ∧xj →∞

P (Xi > xi |Xj > xj ) = 0

for all 1 ≤ i ̸= j < ∞.

If we change the above relation into lim

xi ∧xj →∞

P (|Xi | > xi |Xj > xj ) = 0

for all 1 ≤ i ̸= j < ∞,

then {Xi , i ≥ 1} are said to be tail asymptotically independent (TAI), where xi ∧ xj = min{xi , xj }. Remark 2.1. The terms ‘‘upper tail asymptotic independence (UTAI)’’ and ‘‘tail asymptotic independence (TAI)’’ are named by Liu et al. (2012). Clearly, the UTAI r.v.s can properly cover the WUOD r.v.s, see Example 3.1 of Liu et al. (2012), and hence can still cover both negatively dependent and positively dependent r.v.s. Also remark that if these r.v.s are nonnegative, the two UTAI and TAI structures are equivalent. Based on the dependence structures introduced above, Liu et al. (2012) showed that in a nonstandard renewal risk model, if the claim sizes {Xi , i ≥ 1} are UTAI r.v.s with common distribution F ∈ L ∩ D , and their inter-arrival times {θi , i ≥ 1} are WLOD r.v.s such that lim gL (n)e−ϵ n = 0

n→∞

(2.3)

holds for some ϵ = ϵ(F ) > 0, and {Xi , i ≥ 1}, {θi , i ≥ 1} and {C (t ), t ≥ 0} are mutually independent, then for any fixed T ∈ Λ, it holds uniformly for all t ∈ ΛT ≡ Λ ∩ [0, T ] that

ψr (x, t ) ∼

t



F (xers )dEN (s).

(2.4)

0

We remark that relation (2.3) as a condition must hold for every ϵ > 0. Please see the constant ϵ = − log(Ee−r pˆ θ1 ) − c we take below (3.7), where r may be close to 0, pˆ depends on the distribution F and θ1 is the claim inter-arrival time. Therefore, relation (2.3) must hold for every ϵ > 0 in order for the obtained results to be valid for such arbitrarily given modelling components. In addition, from the above discussion, we find out that the constant ϵ in (2.3) not only relates to the distribution F , but also depends on the distribution of the claim inter-arrival time θ1 . Motivated by the references cited above, in this paper we will further discuss the following issues. (1) We will consider that the claim arrival process {N (t ), t ≥ 0} is a general counting process, which means that neither independence, nor a special dependence structure, is required among {θi , i ≥ 1}. (2) We will deal with the case when the claim sizes {Xi , i ≥ 1} are non-identically distributed r.v.s. (3) We will extend the uniformity of the finite-time ruin probability to the whole time-interval Λ. (4) We will discuss the case when the premium process {C (t ), t ≥ 0} is not necessarily independent of the other sources of randomness. In the following, we state our main results of this paper, among which the first one is concerned with the local uniformity of the finite-time ruin probability with UTAI, non-identically distributed claim sizes and a general claim-arrival process.

1530

Q. Gao, X. Liu / Statistics and Probability Letters 83 (2013) 1527–1538

Theorem 2.1. Consider the nonstandard risk model introduced in Section 1 with r ≥ 0, in which the claim sizes {Xi , i ≥ 1} are UTAI r.v.s with distributions Fi , i ≥ 1, respectively, and for any fixed t > 0, the general claim-arrival process {N (t ), t ≥ 0} satisfies E (N (t ))p+1 < ∞ for some p > JF+ . Assume that there exists a distribution F ∈ L ∩ D such that 0 < S := lim inf inf x→∞ i≥1

Fi (x) F ( x)

≤ lim sup sup x→∞

i ≥1

Fi (x) F (x)

=: M < ∞.

(2.5)

Then for any fixed T ∈ Λ, it holds uniformly for all t ∈ ΛT that t



F (xe )dEN (s) . ψr (x, t ) . M rs

S 0

t



F (xers )dEN (s),

(2.6)

0

if {Xi , i ≥ 1}, {N (t ), t ≥ 0} and {C (t ), t ≥ 0} are mutually independent. Particularly, if Fi ≡ F , i ≥ 1, then relation (2.4) holds uniformly for all t ∈ ΛT . In comparison to Theorem 2.1, the second main result discusses the other case that {C (t ); t ≥ 0} is not necessarily independent of {Xi , i ≥ 1} or {N (t ), t ≥ 0}. Theorem 2.2. Let the conditions of Theorem 2.1 be valid, and further assume F ∈ C . Then relation (2.6) still holds uniformly for all t ∈ ΛT for any fixed T ∈ Λ, if for any fixed t > 0, P ( C (t ) > x) = o(1)F (x),

(2.7)

where  C (t ) is defined in (1.3). Particularly, if Fi ≡ F , i ≥ 1, then relation (2.4) still holds uniformly for all t ∈ ΛT . In the third main result, we extend the uniformity of (2.4) and (2.6) to the whole set Λ. For notational convenience, we denote by G the common distribution of {θi , i ≥ 1}. Theorem 2.3. Under the conditions of Theorem 2.2 with r > 0 and JF− > 0, we further assume that the claim inter-arrival times {θi , i ≥ 1} are WLOD r.v.s such that (2.3) holds for every ϵ > 0, depending on F and G, and that the total discounted amount of premiums is finite, that is, ∞



e−rs C (ds) < ∞ a.s.

0 ≤ C = 0

Then relation (2.6) holds uniformly for all t ∈ Λ, if one of the following conditions is true: (1) the premium process {C (t ), t ≥ 0} is independent of {Xi , i ≥ 1} and {N (t ), t ≥ 0}; (2) the total discounted amount of premiums satisfies P ( C > x) = o(1)F (x). Particularly, if Fi ≡ F , i ≥ 1, then relation (2.4) still holds uniformly for all t ∈ Λ. Finally, we study the tail probability of the discounted aggregate claims, namely P (Dr (t ) > x), which has the same uniform asymptotics as the finite-time ruin probability in the same time-interval, and then can play an important role in proving the first three theorems. Theorem 2.4. Under the conditions of Theorem 2.1, it holds uniformly for all t ∈ ΛT for any fixed T ∈ Λ that t



F (xers )dEN (s) . P (Dr (t ) > x) . M

S 0

t



F (xers )dEN (s).

(2.8)

0

If Fi ≡ F , i ≥ 1, then it holds uniformly for all t ∈ ΛT that P (Dr (t ) > x) ∼

t



F (xers )dEN (s).

(2.9)

0

Also, if the conditions of Theorem 2.3 are valid, then relation (2.8) holds uniformly for all t ∈ Λ. Furthermore, if Fi ≡ F , i ≥ 1, then (2.9) holds uniformly for all t ∈ Λ. In the rest of this paper, we will prepare some lemmas in Section 3 and then prove the main results in Section 4. 3. Lemmas Before proving our main results, we will present some lemmas, among which the first lemma is from Proposition 2.2.1 of Bingham et al. (1987) and Lemma 3.5 of Tang and Tsitsiashvili (2003).

Q. Gao, X. Liu / Statistics and Probability Letters 83 (2013) 1527–1538

1531

Lemma 3.1. If F ∈ D , then (1) for any 0 < pˆ < JF− ≤ JF+ < p < ∞, there exist positive constants Ci and Di , i = 1, 2, such that F (y) F (x)

≥ C1 (x/y)pˆ for all x ≥ y ≥ D1 ,

(3.1)

≤ C2 (x/y)p for all x ≥ y ≥ D2 ;

(3.2)

and F (y) F (x)

(2) for any p > JF+ , it holds that x−p = o(1)F (x). The lemma below is a direct consequence of Theorem 3.3(iv) of Cline and Samorodnitsky (1994) and Lemma 2.5 of Wang et al. (2005). Lemma 3.2. Let X be a r.v. with distribution F ∈ D , and Y be a nonnegative r.v., independent of X . If EY p < ∞ for some p > JF+ , then F (x) ≍ P (XY > x). In the third lemma, we extend Lemma 2.1 of Liu et al. (2012) to the case when the underlying r.v.s are not necessarily identically distributed. Lemma 3.3. Let {Xi , 1 ≤ i ≤ n} be n TAI and real-valued r.v.s with distributions Fi ∈ L ∩ D , 1 ≤ i ≤ n, respectively. Then for any fixed 0 < a ≤ b < ∞,

 P

n 

 ci Xi > x

∼

i=1

n 

P (ci Xi > x)

(3.3)

i =1

holds uniformly for all (c1 , c2 , . . . , cn ) ∈ [a, b]n . Proof. Follow the proof of Lemma 2.1 of Liu et al. (2012) with some slight changes, or see Lemma 3.1(i) of Chen et al. (2013).  The following lemma starts from and can extend Lemma 3.5 of Wang (2008). Lemma 3.4. For the nonstandard risk model introduced in Section 1 with a general claim-arrival process, if the claim sizes {Xi , i ≥ 1} are non-identically distributed by Fi , i ≥ 1, respectively, such that (2.5) holds, then for t ∈ Λ, ∞ 

t



F (xers )dEN (s) .

S 0

P (Xi e−r τi 1{τi ≤t } > x) . M

t



F (xers )dEN (s).

(3.4)

0

i =1

If {Xi , i ≥ 1} are identically distributed by F , then ∞ 

P (Xi e−r τi 1{τi ≤t } > x) =

t



F (xers )dEN (s). 0

i =1

Proof. This proof can be given by going along the same lines of the proof of Lemma 3.5 of Wang (2008) with obvious modifications only.  Lemma 3.5. Under the conditions of Theorem 2.3, we have P (Dr (∞) > x) . M



∞

F (xers )dEN (s).

(3.5)

0

Proof. For any fixed integer n ≥ 1, we write ∆n =

∞

i=n+1

Xi e−r τi . As done by Chen and Ng (2007) (or Tang and Tsitsiashvili,

2004), we will show in a different method that P (∆n > x) is asymptotically negligible in comparison to F (x) as x and n are sufficiently large, where it should be mentioned that the method we used is inspired by Yi et al. (2011). We note that



∞ 

P (∆n > x) ≤ P

 ( Xi e

−r τ i



> x) + P ∆ n > x,

i=n+1

≤

∞  i=n+1

 (Xi e

−r τ i

≤ x)

i=n+1

 P ( Xi e

∞ 

−r τ i

> x) + P

∞  i=n+1

 Xi e

−r τ i

1{Xi e−r τi ≤x} > x

= H1 + H2 .

(3.6)

1532

Q. Gao, X. Liu / Statistics and Probability Letters 83 (2013) 1527–1538

For H1 , it follows from (2.5), (3.1) and (2.2) that for all x ≥ D1 , H1 =

∞   i=n+1

1

∞ 

Fi (x/y)dP (e−r τi ≤ y) . C1−1 MF (x)

0

gL (i)(Ee−r pˆ θ1 )i .

(3.7)

i=n+1

Using (2.3) and taking ϵ = − log(Ee−r pˆ θ1 ) − c for some c > 0, one can easily see that for some integer i0 > 0 such that gL (i) ≤ e−ci (Ee−r pˆ θ1 )−i for all i ≥ i0 , which implies that ∞ 

i0 

gL (i)(Ee−r pˆ θ1 )i ≤

gL (i)(Ee−r pˆ θ1 )i +

e−ci < ∞.

(3.8)

i=i0 +1

i=1

i=1

∞ 

Then for any given ε > 0, there exists some positive integer n0 such that for all n ≥ n0 and all x ≥ D1 , H1 ≤ C1−1 M ε F (x).

(3.9)

For H2 , by Markov’s inequality, it holds that for p > JF as in (3.2), +

 −p

H2 ≤ x

∞ 

E

p Xi e

−r τ i

.

1{Xi e−r τi ≤x}

i=n+1

When 0 < JF+ < 1, we apply the Cr-inequality to have that for 0 < p ≤ 1, ∞ 

H2 ≤ x−p

E (Xi e−r τi )p 1{Xi e−r τi ≤x}

i=n+1

∞  

≤ x −p

i=n+1

∞

xers



0

pe−rsp yp−1 Fi (y)dydP (τi ≤ s).

(3.10)

0

ˆ 2 > 0 such that Similarly to (3.2), for some δ > 0 such that p − δ > JF+ , there are Cˆ 2 > 1 and D F (y) F (x)

≤ Cˆ 2 (x/y)p−δ for all x ≥ y ≥ Dˆ 2 .

(3.11)

ˆ 2 }. If xers < D, then for all i ≥ n + 1, Set D = max{D2 , D xers



pe−rsp yp−1 Fi (y)dy ≤ xp ≤ xp

F (xers )

0

F (D)

.

(3.12)

If xers ≥ D, it follows that for all i ≥ n + 1 and all x ≥ D, xers



−rsp p−1

pe

y

Fi (y)dy =



D

xers



 pe−rsp yp−1 Fi (y)dy = H3 + H4 .

+ 0

0

(3.13)

D

By (3.2), we find that H3 ≤ e−rsp Dp ≤

C2 F (D)

xp F (xers ).

(3.14)

By (2.5) and (3.11), we get xers

 H4 . M

pCˆ 2 xp−δ e−rsδ yδ−1 F (xers )dy ≤

pCˆ 2 M

δ

D

xp F (xers ).

(3.15)

From (3.13)–(3.15), we have that when xers ≥ D,



xers



−rsp p−1

pe 0

y

Fi (y)dy .

C2 F (D)

+

pCˆ 2 M

δ

 xp F (xers ).



Hence, combining (3.12) and (3.16) and setting C = max

1 F (D)

(3.16)

, FC(D2 ) +

pCˆ 2 M

δ



can show that for all i ≥ n + 1 and all x ≥ D,

xers



pe−rsp yp−1 Fi (y)dy ≤ Cxp F (xers ). 0

(3.17)

Q. Gao, X. Liu / Statistics and Probability Letters 83 (2013) 1527–1538

1533

Then, substituting (3.17) into (3.10) and arguing as (3.9), we deduce that for all large n ≥ n0 and all x ≥ max{D1 , D}, ∞ 

H2 ≤ C

P (Xe−r τi > x) ≤ CC1−1 ε F (x),

(3.18)

i=n+1

where X is a r.v. distributed by F . When JF+ > 1, by Minkowski’s inequality and along with the similar lines of the proof of the case that 0 < JF+ < 1, we also obtain that for some constant C > 0 and all x ≥ max{D1 , D},

 H2 ≤ x

∞  

−r τ i p

) 1{Xi e−r τi ≤x}

E (Xi e

−p

 1p

p



P (Xe

≤C

i=n+1

 = C

∞  

−r τ i

1 > x) p

p

i=n+1

∞  

1

F (x/y)dP (e−r τi

    1p p ∞  i  1p p   − 1 ˆ − r p θ 1 ≤ y)  ≤ CC1 F (x) gL (i) Ee ,

0

i=n+1

i=n+1

where the last step is similar to the derivation of (3.7). Arguing as (3.8) still proves that ∞  

gL (i)(Ee−r pˆ θ1 )i

 1p

< ∞.

i =1

Therefore, for all large n ≥ n0 and all x ≥ max{D1 , D}, we also have H2 ≤ CC1−1 ε F (x).

(3.19)

Consequently, from (3.6), (3.9), (3.18), (3.19) and the arbitrariness of ε > 0, it holds for some positive integer n0 that P ∆n0 > x = o(1)F (x).





(3.20)

Let n0 be fixed as above. Note that for any 0 < v < 1,

 P (Dr (∞) > x) ≤ P

n0 

 Xi e

−r τi

  > (1 − v)x + P ∆n0 > v x = H5 + H6 .

(3.21)

i =1

For H5 , applying Theorem 1 of Chen et al. (2010) and relation (2.5), we have

H5 ∼

n0  

1

 Fi

(1 − v)x y

0

i =1

F

≤ M sup



(1−v)x

n0  

dP (e−r τi ≤ y) . M

i =1



y

n0  

F (x/y)

0


i =1

1

1

F



(1−v)x y

F (x/y)

0

 F (x/y)dP (e−r τi ≤ y)

F (x/y)dP (e−r τi ≤ y).

0

Thus, H5

lim sup x→∞

F

M

n0   1 i =1

0

≤ lim sup sup

F (x/y)dP (e−r τi ≤ y)

x→∞

0


(1−v)x y

F (x/y)

 = lim sup x→∞

F ((1 − v)x) F (x)

.

Letting v ↘ 0 and using F ∈ C , it follows that H5 . M

n0 

P (Xe−r τi > x).

(3.22)

i =1

For H6 , by (3.20), F ∈ C ⊂ D and Lemma 3.2, we obtain H6 = o(1)P (Xe−r τ1 > x).

(3.23)

Hence, substituting (3.22) and (3.23) into (3.21), we get that P (Dr (∞) > x) . M

∞ 

P (Xe−r τi > x) = M

F (xers )dEN (s), 0

i=1

which gives the proof of this lemma.

∞





1534

Q. Gao, X. Liu / Statistics and Probability Letters 83 (2013) 1527–1538

The last lemma is due to Lemma 3.3 of Gao et al. (in press). Lemma 3.6. Consider the counting process {N (t ), t ≥ 0} with WLOD inter-arrival times {θi , i ≥ 1} satisfying (2.3) for every ϵ > 0, depending on F and G. Then for any fixed t > 0 and any p > 0, E (N (t ))p < ∞. 4. Proofs of main results Now we are ready to prove the main results of this paper. First of all, we should give the proof of Theorem 2.4 which is helpful for proving Theorems 2.1–2.3. Proof of Theorem 2.4. We follow the ideas as that in the proof of Theorem 1.1 of Tang (2007), but in a simpler manner, to prove the theorem. Firstly, we deal with the uniformity of (2.8) for all t ∈ ΛT under the conditions of Theorem 2.1. For an arbitrarily fixed integer m, we have



∞ 

P (Dr (t ) > x) =

+

n=m+1

  m n   P

n=1

 Xi e

−r τ i

> x, N (t ) = n = K1 + K2 .

(4.1)

i=1

As the division in the proof of Theorem 3.4 of Chen and Yuen (2009), we divide K1 into two parts as

 

 

K1 ≤



+

P

x/D2
m
n 

 Xi > x P (N (t ) = n) = K11 + K12 .

(4.2)

i=1

For K11 , by F ∈ L ∩ D , (2.5) and (3.2), it holds for all large x ≥ D2 and all t ∈ ΛT that n 



K11 ≤



Fi (x/n)P (N (t ) = n) . C2 MF (x)

m
np+1 P (N (t ) = n)

m
≤ C2 MF (x)E (N (T ))

p+1

1{N (T )>m} .

(4.3)

For K12 , by Markov’s inequality, we attain that for all t ∈ ΛT , K12 ≤ P (N (t ) > x/D2 ) ≤ (x/D2 )−p−1 E (N (T ))p+1 1{N (T )>x/D2 } .

(4.4)

Then, from (4.2)–(4.4), Lemma 3.1(2) and the condition on {N (t ), t ≥ 0}, it follows that lim lim sup

m→∞ x→∞ t ∈Λ

T

K1

= 0,

F (x)

which, along with Lemma 3.2, can yield that uniformly for all t ∈ ΛT , K1 = o(1)P (X1 e−r τ1 1{τ1 ≤t } > x) = o(1)

∞ 

P (Xi e−r τi 1{τi ≤t } > x),

as m → ∞ and x → ∞.

(4.5)

i=1

Now consider K2 , by Lemma 3.3 and independence between {Xi , i ≥ 1} and {N (t ), t ≥ 0}, it holds uniformly for all t ∈ ΛT and 1 ≤ n ≤ m that

 P

n 

 Xi e

−r τ i

> x, N (t ) = n



 =

P {0≤t1 ≤···≤tn ≤t , tn+1 >t }

i=1

∼

n  

{0≤t1 ≤···≤tn ≤t , tn+1 >t }

i=1

=

n  

n 

 Xi e

−rti

> x dG(t1 , . . . , tn+1 )

i=1

P (Xi e−rti > x)dG(t1 , . . . , tn+1 )

P Xi e−r τi > x, N (t ) = n ,



i=1

where G(t1 , . . . , tn+1 ) is the joint distribution of random vector (τ1 , . . . , τn+1 ), 1 ≤ n ≤ m. Hence we have that, uniformly for all t ∈ ΛT , K2 ∼

m  n  

P Xi e−r τi > x, N (t ) = n



n=1 i=1

 =

∞  n =1

−

∞ 



n=m+1

n  

P Xi e−r τi > x, N (t ) = n = K21 − K22 .

i=1



(4.6)

Q. Gao, X. Liu / Statistics and Probability Letters 83 (2013) 1527–1538

1535

For K21 , it is clear that K21 =

∞  ∞ 

P (Xi e−r τi > x, N (t ) = n) =

i=1 n=i

∞ 

P (Xi e−r τi 1{τi ≤t } > x).

(4.7)

i =1

For K22 , we know that for all t ∈ ΛT , K22 . MF (x)

∞ 

nP (N (t ) = n) ≤ MF (x)EN (T )1{N (T )>m} .

n=m+1

Similarly to the derivation for K1 , we still get that uniformly for all t ∈ ΛT , K22 = o(1)

∞ 

P (Xi e−r τi 1{τi ≤t } > x),

as m → ∞ and x → ∞.

(4.8)

i=1

From these arguments above, we obtain that uniformly for all t ∈ ΛT , P (Dr (t ) > x) ∼

∞ 

P (Xi e−r τi 1{τi ≤t } > x).

(4.9)

i =1

Thus by Lemma 3.4, we can prove that relation (2.8) holds uniformly for all t ∈ ΛT . If Fi ≡ F , i ≥ 1, then S = M = 1. So from (2.8), we easily obtain the uniformity of (2.9) for all t ∈ ΛT . Subsequently, we extend the interval in which (2.8) and (2.9) hold uniformly to an infinite interval Λ under the conditions of Theorem 2.3. Clearly, the uniformity of (2.8) for all t ∈ ΛT0 follows immediately from the first half of this proof, where T0 ∈ Λ is an arbitrarily fixed number. Below we turn to prove that (2.8) holds uniformly for all t ∈ (T0 , ∞]. Again by (3.1), it holds for all t ∈ Λ and all x ≥ D1 that

∞ t t 0

F (xers )dEN (s)

F (xers )dEN (s)

∞ = t t

F (xers )/F (xert )dEN (s)

F (xers )/F (xert )dEN (s)

0

∞ ≤ C1−2 t t 0

e−r pˆ s dEN (s)

e−r pˆ s dEN (s)

.

(4.10)

By (2.2), it follows that ∞



e−r pˆ s dEN (s) = 0

∞  

∞

e−r pˆ s dP (τn ≤ s) = 0

n =1

∞ 

E (e−r pˆ τn ) ≤

n=1

∞ 

gL (n)(Ee−r pˆ θ1 )n < ∞,

n =1

where the last step is similar to the derivation of (3.8). Hence, the right-hand side of (4.10) tends to 0 as t tends to ∞. Thus for arbitrarily fixed ε > 0, there exists some large T0 ∈ Λ such that for all x ≥ D1 ,



∞

F (xers )dEN (s) ≤ ε

T0



T0

F (xers )dEN (s).

(4.11)

0

By Lemma 3.5 and (4.11), it holds uniformly for all t ∈ (T0 , ∞] that P (Dr (t ) > x) ≤ P (Dr (∞) > x) . M

∞



F (xers )dEN (s) 0

t





≤M

∞

+ 0

F (xers )dEN (s) ≤ (1 + ε)M

T0

t



F (xers )dEN (s).

(4.12)

0

Likewise, by (2.8) with t replaced by T0 and (4.11), we attain that uniformly for all t ∈ (T0 , ∞], P (Dr (t ) > x) ≥ P (Dr (T0 ) > x) & S

T0



F (xers )dEN (s)

0

≥

S 1+ε



∞

F (xers )dEN (s) ≥

0

S

t



1+ε

F (xers )dEN (s).

(4.13)

0

Consequently, by (4.12), (4.13) and the arbitrariness of ε > 0, we see that relation (2.8) holds uniformly for all t ∈ (T0 , ∞]. Particularly, if Fi ≡ F , i ≥ 1, then relation (2.9) still holds uniformly for all t ∈ Λ because of the uniformity of (2.8) with S = M = 1.  Proof of Theorem 2.1. By (1.1) and (1.4), we have

  ψr (x, t ) = P Dr (s) > x +  C (s) for some 0 < s ≤ t ,

1536

Q. Gao, X. Liu / Statistics and Probability Letters 83 (2013) 1527–1538

where Dr (s) and  C (s) are those as in (1.2) and (1.3), respectively. Then, it is easy to obtain that for any t ∈ Λ,

ψr (x, t ) ≤ P (Dr (t ) > x)

(4.14)

and





ψr (x, t ) = P

    Dr (s) > x +  C ( s) ≥ P Dr (t ) > x +  C (t ) .

(4.15)

0
On the one hand, from (4.14) and the uniformity of (2.8) for all t ∈ ΛT , it follows immediately that, uniformly for all t ∈ ΛT ,

ψr (x, t ) . M

t



F (xers )dEN (s).

(4.16)

0

On the other hand, by (4.15), the uniformity of (2.8) for all t ∈ ΛT , F ∈ L ∩ D , and independence between {C (t ), t ≥ 0} and the other sources of randomness, it holds uniformly for all t ∈ ΛT that

 ∞   P (Dr (t ) > x + y)P ( C (T ) ∈ dy) ψr (x, t ) ≥ P Dr (t ) > x + C (T ) = 0  ∞ t & S F ((x + y)ers )dEN (s)P ( C (T ) ∈ dy) 

0

0 t



F (xers )dEN (s),

∼S

(4.17)

0

where in the last step we applied the local uniformity of (2.1). Hence, combining (4.16) and (4.17) can imply that (2.6) holds uniformly for all t ∈ ΛT . Particularly, if Fi ≡ F , i ≥ 1, one has S = M = 1, and then (2.4) holds uniformly for all t ∈ ΛT .  Proof of Theorem 2.2. Similarly to the proof of Theorem 2.1, it remains to show the uniform asymptotic lower-bound of

ψr (x, t ) for all t ∈ ΛT under the conditions of this theorem. Note that F ∈ C , one can know that for any given ε > 0, there exists some δ0 > 0 such that for all large x, F ((1 + δ0 )x) ≥ (1 − ε)F (x).

(4.18)

It follows from (4.15) that for δ0 > 0 as above and all t ∈ ΛT ,

ψr (x, t ) ≥ P (Dr (t ) > x +  C (T )) ≥ P (Dr (t ) > (1 + δ0 )x) − P ( C (T ) > δ0 x) = J1 − J2 .

(4.19)

For J1 , by the uniformity of (2.8) for all t ∈ ΛT and (4.18), it holds uniformly for all t ∈ ΛT and all large x that t



F ((1 + δ0 )xers ) dEN (s) ≥ (1 − ε)S

J1 & S

t



0

F (xers ) dEN (s).

(4.20)

0

For J2 , by (2.7) and F ∈ C ⊂ D , we find lim sup x→∞

J2 F (x)

= lim sup x→∞

J2 F (δ0 x)

·

F (δ0 x) F (x)

= 0.

This, along with Lemma 3.2, yields that for all large x and all t ∈ ΛT , J2 ≤ ε P (X1 e−r τ1 1{τ1 ≤t } > x) ≤ ε

∞ 

P (Xi e−r τi 1{τi ≤t } > x)

i=1

.

Mε S

t



F (xers )dEN (s),

·S

(4.21)

0

where the last step is from Lemma 3.4. Thus, substituting (4.20) and (4.21) into (4.19) and using the arbitrariness of ε > 0 can show the uniform asymptotic lower-bound of ψr (x, t ) for all t ∈ ΛT , and so complete the proof.  Proof of Theorem 2.3. Following Theorems 2.1–2.2 and Lemma 3.6, we can easily know that relation (2.6) holds uniformly for all t ∈ ΛT0 under conditions 1 and 2 of Theorem 2.3. Hence, we will achieve the proof if we prove that relation (2.6) still holds uniformly for all t ∈ (T0 , ∞]. For the uniform asymptotic upper-bound for ψr (x, t ), again by (4.14) and Theorem 2.4, it holds uniformly for all t ∈ Λ that

ψr (x, t ) . M

t



F (xers )dEN (s). 0

Q. Gao, X. Liu / Statistics and Probability Letters 83 (2013) 1527–1538

1537

Hence, we only need to show the corresponding uniform asymptotic lower-bound of ψr (x, t ) for all t ∈ (T0 , ∞], that is,

ψr (x, t ) & S

t



F (xers )dEN (s)

(4.22)

0

holds uniformly for all t ∈ (T0 , ∞]. In what follows, we formulate the proof into two parts. In the first part, we deal with the case of condition 1. By (4.15), (2.8) with t = T0 , F ∈ C ⊂ L and the local uniformity of (2.1), we prove that uniformly for all t ∈ (T0 , ∞],

 ψ(x, t ) ≥ P Dr (T0 ) > x +  C = 

∞



P (Dr (T0 ) > x + y)P ( C ∈ dy) 0

∞



T0



& S 0

0

1+ε

T0



F (xers )dEN (s)

0 t



S

≥

F ((x + y)e )dEN (s)P ( C ∈ dy) ∼ S rs

F (xers )dEN (s), 0

where the last step is due to (4.11). Because ε > 0 is arbitrary, the relation (4.22) holds uniformly for all t ∈ (T0 , ∞]. In the second part, we turn to the case of condition 2. By (4.15), we have that for the fixed δ0 > 0 as in (4.18) and all t ∈ (T0 , ∞],

    ψr (x, t ) ≥ P Dr (T0 ) > x +  C ≥ P Dr (T0 ) > (1 + δ0 )x − P ( C > δ0 x) = J3 − J4 .

(4.23)

For J3 , by (2.8) with t = T0 , (4.18) and (4.11), it holds uniformly for all t ∈ (T0 , ∞] and all large x that T0

 J3 & S

F ((1 + δ0 )xe )dEN (s) ≥ (1 − ε)S rs



T0

F (xers )dEN (s)

0

0

≥ (1 − 2ε)S



t

F (xers )dEN (s).

(4.24)

0

For J4 , by condition 2 and F ∈ C ⊂ D , we attain that for all t ∈ (T0 , ∞], lim sup  t x→∞

0

J4 F(

xers

)dEN (s)

≤ lim sup  T x→∞

≤ lim sup x→∞

0

0

J4 F (xers )dEN (s)

J4 F (δ0 x)

·

F (δ0 x) F (xerT0 )EN (T0 )

= 0,

which implies that for all t ∈ (T0 , ∞] and all large x, J4 ≤ ε

t



F (xers )dEN (s).

(4.25)

0

Consequently, by (4.23)–(4.25) and the arbitrariness of ε > 0, we obtain the uniformity of (4.22) for all t ∈ (T0 , ∞]. For the particular case that Fi ≡ F , i ≥ 1, the uniformity of (2.4) for all t ∈ Λ is easily obtained by (2.6) with S = M = 1.  Acknowledgements The authors would like to thank the Editor and an anonymous referee for their very valuable comments on an earlier version of the paper. This research was supported by the Research Start-up Funding for PhD of Nanjing Audit University (No. NSRC10022). References Asimit, A.V., Badescu, A.L., 2010. Extremes on the discounted aggregate claims in a time dependent risk model. Scand. Actuar. J. (2), 93–104. Bingham, N.H., Goldie, C.M., Teugels, J.L., 1987. Regular Variation. Cambridge Unversity Press, Cambridge. Chen, Y., Ng, K.W., 2007. The ruin probability of the renewal model with constant interest force and negatively dependent heavy-tailed claims. Insur. Math. Econ. 40, 415–423. Chen, Y., Wang, L., Wang, Y., 2013. Uniform asymptotics for the finite-time ruin probabilities of two kinds of nonstandard bidimensional risk models. J. Math. Anal. Appl. 401, 114–129. Chen, Y., Yuen, K., 2009. Sums of pairwise quasi-asymptotically independent random variables with consistent variation. Stoch. Models 25, 76–89. Chen, Y., Yuen, K., 2012. Precise large deviations of aggregate claims in a size-dependent renewal risk model. Insur. Math. Econ. 51, 457–461. Chen, Y., Zhang, W., Liu, J., 2010. Asymptotic tail probability of randomly weighted sum of dependent heavy-tailed random variables. Asia-Pac. J. Risk Insur. 4 (2). http://dx.doi.org/10.2202/2153-3792.1055. Article 4.

1538

Q. Gao, X. Liu / Statistics and Probability Letters 83 (2013) 1527–1538

Cline, D.B.H., Samorodnitsky, G., 1994. Subexponentiality of the product of independent random variables. Stoch. Proc. Appli. 49, 75–98. Embrechts, P., Klüppelberg, C., Mikosch, T., 1997. Modelling Extremal Events for Insurance and Finance. Springer, Berlin. Gao, Q., Jin, N., Gu, P., 2012. Asymptotic behavior of the finite-time ruin probability with pairwise quasi-asymptotically independent claims and constant interest force. Rocky Mountain J. Math. (in press). Geluk, J., Tang, Q., 2009. Asymptotic tail probabilities of sums of dependent subexponential random variables. J. Theoret. Probab. 22, 871–882. Hao, X., Tang, Q., 2008. A uniform asymptotic estimate for discounted aggregate claims with sunexponential tails. Insur. Math. Econ. 43, 116–120. Li, J., Tang, Q., Wu, R., 2010. Subexponential tails of discounted aggregate claims in a time-dependent renewal risk model. Adv. Appl. Probab. 42, 1126–1146. Li, J., Wang, K., Wang, Y., 2009. Finite-time ruin probability with NQD dominated varying-tailed claims and NLOD inter-arrival times. J. Syst. Sci. Complex. 22, 407–414. Liu, X., Gao, Q., Wang, Y., 2012. A note on a dependent risk model with constant interest rate. Statist. Probab. Lett. 82, 707–712. Rolski, T., Schmidli, H., Schmidt, V., Teugels, J., 1999. Stochastic Processes for Insurance and Finance. John Wiley & Sons. Tang, Q., 2005. Asymptotic ruin probabilities of the renewal model with constant interest force and regular variation. Scand. Actuar. J. (1), 1–5. Tang, Q., 2007. Heavy tails of discounted aggregate claims in the continous-time renewal model. J. Appl. Probab. 44, 285–294. Tang, Q., Tsitsiashvili, G., 2003. Precise estimates for the ruin probability in finite horizon in a discrete-time model with heavy-tailed insurance and financial risks. Stoch. Proc. Appl. 108, 299–325. Tang, Q., Tsitsiashvili, G., 2004. Finite- and infinite-time ruin probabilities in the presence of stochastic returns on investments. Adv. Appl. Probab. 36 (4), 1278–1299. Wang, D., 2008. Finite-time probabiliy with heavy-tailed claims and constant interest rate. Stoch. Models 24, 41–57. Wang, D., Su, C., Zeng, C., 2005. Uniform estimate for maximum of randomly weighted sums with applications to insurance risk theory. Sci. China A 48, 1379–1394. Wang, K., Wang, Y., Gao, Q., 2013. Uniform asymptotics for the finite-time ruin probability of a dependent risk model with a constant interest rate. Methodol. Comput. Appl. Probab. 15, 109–124. Yang, Y., Wang, Y., 2010. Asymptotics for ruin probability of some negatively dependent risk models with a constant interest rate and dominatedly-varyingtailed claims. Statist. Probab. Lett. 80, 143–154. Yi, L., Chen, Y., Su, C., 2011. Approximation of the tail probability of randomly weighted sums of dependent random variables with dominated variation. J. Math. Anal. Appl. 376, 365–372.