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Research article

Estimations for aggregate amount of claims in a risk model with arbitrary dependence between claim sizes and inter-arrival times

  • Received: 31 May 2022 Revised: 21 July 2022 Accepted: 27 July 2022 Published: 02 August 2022
  • MSC : 60F10, 91B05, 91G05

  • This paper considers a compound risk model, in which the individual claim sizes and their inter-arrival times can be arbitrarily dependent. We mainly investigate the claim sizes are extended negatively dependent. When the claim sizes have consistently-varying-tailed distributions, we obtain precise large deviations of the aggregate amount of claims in the above dependent compound risk model.

    Citation: Weiwei Ni, Chenghao Xu, Kaiyong Wang. Estimations for aggregate amount of claims in a risk model with arbitrary dependence between claim sizes and inter-arrival times[J]. AIMS Mathematics, 2022, 7(10): 17737-17746. doi: 10.3934/math.2022976

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  • This paper considers a compound risk model, in which the individual claim sizes and their inter-arrival times can be arbitrarily dependent. We mainly investigate the claim sizes are extended negatively dependent. When the claim sizes have consistently-varying-tailed distributions, we obtain precise large deviations of the aggregate amount of claims in the above dependent compound risk model.



    In this paper, we consider a compound renewal risk model. Let the individual claim sizes {Xi,i1} be a sequence of dependent nonnegative random variables (r.v.s) with common distribution FX and finite mean μX>0 and the inter-arrival times of events {θi,i1} be a sequence of independent and identically distributed (i.i.d.) nonnegative r.v.s with common distribution Fθ and finite mean β1>0. Let Zn be the number of claims caused by the nth event, n1. Assume that {Zi,i1} is a sequence of i.i.d. positive integer r.v.s with common distribution FZ and finite mean μZ>0. Suppose that {Zk,k1} are independent of {Xk,k1} and {θk,k1}, and {Xk,k1} and {θk,k1} may be arbitrarily dependent. The number of events up to time t0 is a renewal counting process

    N(t)=sup{n1,ni=1θit}. (1.1)

    Let θ(t)=E(N(t)), t0. Since {θk,k1} are i.i.d., by the elementary renewal theorem we get θ(t)/tβ as t. The number of claims up to time t0 is a compound renewal counting process

    Λ(t)=N(t)k=1Zk. (1.2)

    Since {Zi,i1} and {θi,i1} are independent, the mean function of {Λ(t),t0} is λ(t)=E(Λ(t))=μZθ(t),t0 and λ(t)/tμZβ as t. Thus, the aggregate amount of claims up to time t0 is denoted by

    SΛ(t)=Λ(t)k=1Xk,t0, (1.3)

    which is called a compound sum.

    For the compound sum SΛ(t),t0, the early studies are focus on the independent structure, where {Xi,i1} is a sequence of i.i.d. r.v.s and independent of {θi,i1}. We refer the reader to Tang et al. [12], Aleškevičienė et al. [1], Konstantinides and Loukissas [9], Yang et al. [17], Wang et al. [14], among others.

    In recent years, more and more researchers are interested in dependent compound renewal risk model. They add some dependent structures on r.v.s {Xi,i1}, {θi,i1} and {Zi,i1}. For example, Yang et al. [16] considered {Xi,i1}, {θi,i1} and {Zi,i1} are extended negatively dependent r.v.s, respectively. Konstantinides and Loukissas [10] and Chen et al. [5] considered the case that {Xi,i1} are negatively dependent r.v.s. Wang and Chen [13] investigated there was a pairwise negatively quadrant dependence structure or the upper tail asymptotical dependence structure in the claim sizes {Xi,i1}. In the above results, they need {Xi,i1} and {θi,i1} are independent.

    However, in order to reflect the insurance practice, some researchers put some dependence structures between {Xi,i1} and {θi,i1}. For the renewal risk model, Chen and Yuen [4] introduced a size-dependent risk model and Bi and Zhang [2] considered a regression-type size-dependent risk model. They obtained some results of precise large deviation of the aggregate amount of claims. Guo et al. [8] and Zhou et al. [18] obtained the precise large deviations for the compound sum SΛ(t),t0 for the dependence structures of Chen and Yuen [4] and Bi and Zhang [2], respectively.

    In this paper, we will still investigate the precise large deviations of SΛ(t),t0. We will drop the independent assumption or dependence structures between the claim sizes {Xi,i1} and the inter-arrival times of events {θi,i1} under the condition ¯Fθ(t)=o(¯FX(t)) as t.

    The rest of the paper is organized as follows. In Section 2, we present some necessary preliminaries and main results. In Section 3 the proofs of main results are given.

    Without special statement, in this paper a limit is taken as t. For a real-valued number a, let a+=max{0,a} and a=min{0,a}. a denotes the integer part of a. For two positive functions a() and b(), we write a(t)b(t) if lim supa(t)/b(t)1; write a(t)b(t) if lim infa(t)/b(t)1; write a(t)b(t) if lima(t)/b(t)=1; and write a(t)=o(b(t)) if lima(t)/b(t) =0. For two positive bivariate functions g(,) and h(,), we write g(x,t)h(x,t) holds uniformly for xΔϕ as t, if

    lim suptsupxΔg(x,t)h(x,t)1;

    write g(x,t)h(x,t) holds uniformly for xΔϕ as t, if

    lim inftinfxΔg(x,t)h(x,t)1;

    write g(x,t)h(x,t) holds uniformly for xΔϕ as t, if

    limtsupxΔ|g(x,t)h(x,t)1|=0.

    Here we recall some subclasses of the heavy-tailed distribution class which we consider in our paper. A distribution V on (,) with the tail ¯V=1V is said to have a dominatedly varying tail, denoted by VD, if

    lim supt¯V(yt)¯V(t)<,

    for any fixed y>0. A slightly smaller class is C. A distribution V on (,) is said to have a consistently varying tail, denoted by VC, if

    limy1lim infx¯V(xy)¯V(x)=limy1lim supx¯V(xy)¯V(x)=1.

    A distribution V on (,) is said to have a long tail, denoted by VL, if

    limx¯V(xy)¯V(x)=1,

    for any fixed y(,). It is well-known that the following inclusion relationships hold:

    CLDL,

    (see, e.g., Cline and Samorodnitsky [3], Embrechts et al. [7] and Xun et al. [15]).

    In this paper, we will consider the claim sizes have the following dependence structure. According to Liu [11], a sequence of real-valued r.v.s ξ1,ξ2, is said to be upper extended negatively dependent UEND if there exists some positive constant M, such that for each k=1,2, and all y1,,yk(,)

    P(ki=1{ξi>yi})Mki=1P(ξi>yi);

    they are said to be lower extended negatively dependent (LEND) if there exists some positive constant M, such that for each k=1,2, and all y1,,yk(,)

    P(ki=1{ξiyi})Mki=1P(ξiyi);

    and the r.v.s ξ1,ξ2, are said to be extended negatively dependent END if they are both UEND and LEND.

    The following is the main result of this paper, which gives the precise large deviations for the compound sum SΛ(t),t0.

    Theorem 2.1. Consider the compound sum (1.3). Assume that the claim sizes {Xi,i1} are END r.v.s with common distribution FXC, E(Xr1)< for some r>1, ¯Fθ(t)=o(¯FX(t)) and ¯FZ(t)=o(¯FX(t)). Then for any 0<γ<Γ<, it holds uniformly for all x[γλ(t),Γλ(t)] that

    P(SΛ(t)μXλ(t)>x)λ(t)¯FX(x). (2.1)

    When Zi1, i1, then SΛ(t)=SN(t), t0. From Theorem 2.1 the following corollary can be obtained.

    Corollary 2.1. Assume that the claim sizes {Xi,i1} are END r.v.s with common distribution FXC, E(Xr1)< for some r>1 and ¯Fθ(t)=o(¯FX(t)). Then for any 0<γ<Γ<, it holds uniformly for all x[γt,Γt] that

    P(SN(t)μXβt>x)βt¯FX(x). (2.2)

    Remark 2.1. When {Xi,i1} are i.i.d. r.v.s, Chen et al. [6] obtained (2.2) under the condition 0<μX<, FxC and ¯Fθ(t)=o(¯FX(t)). Thus Corollary 2.1 extends the result of Chen et al. [6] to the dependent claim sizes {Xi,i1}.

    Before giving the proof of Theorem 2.1, we first give two lemmas. The following lemma comes from Theorem 2.1 of Liu [11].

    Lemma 3.1. Let {ξi,i1} be a sequence of END nonnegative r.v.s with common distribution VC and finite mean μ>0. If E(ξ1)r< for some r>1, then for any γ>0

    P(ni=1ξinμ>x)n¯V(x)

    holds uniformly for all xγn as n.

    The following lemma is Corollary 3.1 of Chen et al. [6].

    Lemma 3.2. Let ξ1,ξ2,... be i.i.d. copies of real-valued r.v. ξ with mean 0. Suppose that P(|ξ|>x)=o(¯V(x)) for some VC. Then for any given γ>0, it holds uniformly for all xγn as n that

    P(|ni=1ξk|>x)=o(n¯V(x)).

    Now we turn to the proof of Theorem 2.1. In the proof, unless otherwise stated, a limit is understood as being valid uniformly for all γλ(t)xΓλ(t) as t for any 0<γ<Γ<.

    Proof of Theorem 2.1. For any 0<γ<Γ<, since λ(t)μZβt, for any small 0<ϵ<max{1βμZ,γμXμZβ}, there exists t0>0 such that for any t>t0,

    |λ(t)tμZβ|<ϵ. (3.1)

    We first prove

    P(SΛ(t)μXλ(t)>x)λ(t)¯FX(x). (3.2)

    For any μZϵβ<δ1<1, it holds for all x>0 and t>0 that

    P(SΛ(t)μXλ(t)>x)P((1δ1)λ(t)k=1XkμXλ(t)>x,Λ(t)(1δ1)λ(t))P((1δ1)λ(t)k=1XkμXλ(t)>x)P(Λ(t)<(1δ1)λ(t))=:I1(x,t)I2(t). (3.3)

    Applying Lemma 3.1, it holds that

    I1(x,t)=P((1δ1)λ(t)k=1Xk(1δ1)λ(t)μX>x+μXλ(t)μX(1δ1)λ(t))(1δ1)λ(t)¯FX(x+μXλ(t)μX(1δ1)λ(t))((1δ1)λ(t)1)¯FX((1+μXδ1γ)x+μX), (3.4)

    where the second step uses the facts γ+δ1μX>0, and for any 0<γ<γ+μXδ11δ1, it holds that x+μXλ(t)μX(1δ1)λ(t)γ(1δ1)λ(t). Since FXCL, we have

    I1(x,t)((1δ1)λ(t)1)¯FX((1+μXδ1γ)x).

    By FXC and letting ϵ0 and δ10 we have

    I1(x,t)λ(t)¯FX(x). (3.5)

    For I2(t), taking μZϵβ<ϵ1<δ1 and then split I2(t) as follows:

    I2(t)=P(N(t)j=1Zj<(1δ1)λ(t),N(t)λ(t)μZ(1ϵ1))+P(N(t)j=1Zj<(1δ1)λ(t),N(t)<λ(t)μZ(1ϵ1))=:K1(t)+K2(t). (3.6)

    For K1(t), it holds that for all x>0 and t>0 that

    K1(t)P(λ(t)μZ(1ϵ1)j=1Zj<(1δ1)λ(t))P(λ(t)μZ(1ϵ1)j=1(ZjμZ)<(ϵ1δ1)λ(t)+μZ).

    By Lemma 3.2 and FCLD it holds that

    K1(t)=o(λ(t)¯FX(λ(t)(δ1ϵ1)μZ))=o(λ(t)¯FX(Γ1(δ1ϵ1)xμZ))=o(λ(t)¯FX(x)), (3.7)

    where in the first step Lemma 3.2 can be used because of for any 0<γ<(δ1ϵ1)μZ1ϵ1, it holds that (δ1ϵ1)λ(t)μZγλ(t)μZ(1ϵ1).

    For K2(t), by Lemma 3.2 and FXC we have

    P(N(t)<λ(t)μZ(1ϵ1))P(λ(t)μZ(1ϵ1)+1j=1θj>t)P(λ(t)μZ(1ϵ1)+1j=1(θj1β)>tλ(t)μZ(1ϵ1)+1β)=o(λ(t)¯FX(tλ(t)βμZ+ϵ1λ(t)βμZ1β))=o(λ(t)¯FX(ϵ1λ(t)βμZ))=o(λ(t)¯FX((ΓβμZ)1ϵ1x))=o(λ(t)¯FX(x)), (3.8)

    where in the third step Lemma 3.2 is used, which is due to the fact that for any 0<ˆγ<ϵ1μZϵββ(1ϵ1), it holds that tλ(t)βμZ+ϵ1λ(t)βμZ1β>ˆγ(λ(t)μZ(1ϵ1)+1).

    By (3.7) and (3.8), we have

    I2(t)=o(λ(t)¯FX(x)). (3.9)

    Using (3.5), (3.9) and (3.3), we know that (3.2) holds.

    Next we will prove

    P(SΛ(t)μXλ(t)>x)λ(t)¯FX(x). (3.10)

    For any fixed μZϵβ<δ2<γμX, it holds for any x>0 and t>0 that

    P(SΛ(t)μXλ(t)>x)=P(SΛ(t)μXλ(t)>x,Λ(t)(1+δ2)λ(t))+P(SΛ(t)μXλ(t)>x,Λ(t)>(1+δ2)λ(t))P((1+δ2)λ(t)k=1XkμXλ(t)>x)+P(Λ(t)>(1+δ2)λ(t))=:J1(x,t)+J2(t). (3.11)

    By Lemma 3.1, we have

    J1(x,t)=P((1+δ2)λ(t)k=1Xk(1+δ2)λ(t)μX>x+μXλ(t)μX(1+δ2)λ(t))(1+δ2)λ(t)¯FX(x+μXλ(t)μX(1+δ2)λ(t))((1+δ2)λ(t))¯FX((1μXδ2γ)x), (3.12)

    where in the second step Lemma 3.1 can be used because of for any 0<˜γ<γμXδ21+δ2, it holds that x+μXλ(t)μX(1+δ2)λ(t)˜γ(1+δ2)λ(t). Since FXCLD, by letting ϵ0 and δ20, it holds that

    J1(x,t)λ(t)¯FX(x). (3.13)

    For J2(t), taking μZϵβ<ϵ2<δ2 it holds for any x>0 and t>0 that

    J2(t)=P(N(t)j=1Zj>(1+δ2)λ(t),N(t)λ(t)μZ(1+ϵ2))+P(N(t)j=1Zj>(1+δ2)λ(t),N(t)>λ(t)μZ(1+ϵ2))=:K3(t)+K4(t). (3.14)

    For K3(t), it holds for any x>0 and t>0 that

    K3(t)P(λ(t)μZ(1+ϵ2)j=1(ZjμZ)>(1+δ2)λ(t)μZλ(t)μZ(1+ϵ2))P(λ(t)μZ(1+ϵ2)j=1(ZjμZ)>(δ2ϵ2)λ(t)).

    Thus

    K3(t)=o(λ(t)¯FX(λ(t)(δ2ϵ2)))=o(λ(t)¯FX(Γ1(δ2ϵ2)x))=o(λ(t)¯FX(x)), (3.15)

    where in the first step Lemma 3.2 can be used because of for any 0<ˉγ<(δ2ϵ2)μZ1+ϵ2, it holds that (δ2ϵ2)λ(t)ˉγλ(t)μZ(1+ϵ2). The last step can be verified by FXCD.

    For K4(t), by Lemma 3.2 and FXC we have

    P(N(t)>λ(t)μZ(1+ϵ2))P(λ(t)μZ(1+ϵ2)j=1θjt)P(λ(t)μZ(1+ϵ2)j=1(θj1β)tλ(t)μZ(1+ϵ2)1β)=o(λ(t)¯FX(λ(t)βμZ+ϵ2λ(t)βμZt1β))=o(λ(t)¯FX(ϵ2λ(t)βμZ))=o(λ(t)¯FX((ΓβμZ)1ϵ2x))=o(λ(t)¯FX(x)), (3.16)

    where in the third step Lemma 3.2 is used, which is due to the fact that for any 0<˘γ<ϵ2μZϵββ(1+ϵ2), it holds that λ(t)μZ(1+ϵ2)1βt˘γ(λ(t)μZ(1+ϵ2)).

    By (3.15) and (3.16), we have

    J2(t)=o(λ(t)¯FX(x)). (3.17)

    Therefore, by (3.13), (3.17) and (3.11), we know that (3.10) holds.

    Proof of Corollary 2.1. By Theorem 2.1 for any 0<γ1<Γ1<, it holds uniformly for x[γ1θ(t),Γ1θ(t)] that

    P(SN(t)μXθ(t)>x)θ(t)¯FX(x). (3.18)

    For any 0<γ<Γ<, since θ(t)βt, taking 0<γ1<γβ and Γβ<Γ1<, when t is sufficiently large, it holds that

    γ1θ(t)<γtxΓt<Γ1θ(t).

    Since θ(t)βt, for any small enough 0<ϵ<1, when t is sufficiently large it holds that

    0<βtϵθ(t)βt+ϵ. (3.19)

    Thus by (3.18), (3.19) and FCL it holds uniformly for x[γt,Γt] that

    P(SN(t)μXβt>x)P(SN(t)μXθ(t)>xμXϵ)θ(t)¯FX(xμXϵ)βt¯FX(x)

    and

    P(SN(t)μXβt>x)P(SN(t)μXθ(t)>x+μXϵ)θ(t)¯FX(x+μXϵ)βt¯FX(x).

    This completes the proof of Corollary 2.1.

    In this paper we use the probability limiting theory to investigate the aggregate amount of claims of a compound risk model. When the claims have heavy-tailed distributions we give the precise large deviations of the aggregate amount of claims. Under some technical conditions we drop the independent assumption or dependence structures between the claim sizes and the inter-arrival times of events, which expands the use range of the main results.

    This work is supported by the 333 High Level Talent Training Project of Jiangsu Province and the Jiangsu Province Key Discipline in the 14th Five-Year Plan. The authors wish to thank the referees for their very valuable comments on an earlier version of this paper.

    The authors declare no conflict of interest.



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