A Note on the Tail Behavior of Randomly Weighted Sums with Convolution-Equivalently Distributed Random Variables

Abstract

We investigate the tailed asymptotic behavior of the randomly weighted sums with increments with convolution-equivalent distributions. Our obtained result can be directly applied to a discrete-time insurance risk model with insurance and financial risks and derive the asymptotics for the finite-time probability of the above risk model.

1. Introduction and Main Result

Let {𝑋_𝑛,𝑛≥1} be a sequence of independent and identically distributed (i.i.d.) real-valued random variables with common distribution 𝐹, and let {𝑌_𝑛,𝑛≥1} be another sequence of i.i.d. nonnegative r.v.s with common distribution 𝐺 and right endpoint 𝑥_𝐺=sup{𝑥∶ℙ(𝑌₁≤𝑥)<1}. Assume that {𝑋_𝑛,𝑛≥1} are independent of {𝑌_𝑛,𝑛≥1}. In this paper, we are interested in the randomly weighted sum 𝑆^𝑌_𝑛=^𝑛_𝑖=1𝑋_𝑖^𝑖_𝑗=1𝑌_𝑗,𝑛≥1. (1) This is because the study for the tail probability ℙ(𝑆^𝑌_𝑛>𝑥) can be directly applied to risk theory. Consider a discrete-time insurance risk model. Within period 𝑖, 𝑖≥1, the net insurance loss is denoted by a real-valued (r.v.) 𝑋_𝑖. The insurer makes both risk-free and risky investments, leading to an overall stochastic discounted factor 𝑌_𝑖 from time 𝑖 to time 𝑖−1. In the terminology of Norberg [1], the sequences {𝑋_𝑛,𝑛≥1} and {𝑌_𝑛,𝑛≥1} are called the insurance and financial risks, respectively. Then, the randomly weighted sum 𝑆^𝑌_𝑛 in (1) represents the stochastic discounted value of aggregate net losses up to time 𝑛, 𝑛≥1. As usual, the probability of ruin by time 𝑛 can be defined by Ψ(𝑥,𝑛)=ℙmax^𝑚_{1≤𝑚≤𝑛𝑖=1}𝑋_𝑖^𝑖_𝑗=1𝑌_𝑗>𝑥,𝑛≥1, (2) where 𝑥≥0 is interpreted as the initial capital reserve of an insurance company. Clearly, for each 𝑛≥1, ℙ𝑆^𝑌_𝑛>𝑥≤Ψ(𝑥,𝑛)≤ℙ^𝑛_𝑖=1𝑋⁺_𝑖^𝑖_𝑗=1𝑌_𝑗>𝑥, (3) where 𝑋⁺_𝑖=𝑋_𝑖𝟏_{{𝑋𝑖≥0}} denotes the positive part of 𝑋_𝑖, 𝑖≥1. If we can establish an asymptotic formula for ℙ(𝑆^𝑌_𝑛>𝑥) while doing so does not require 𝐹(0−)>0, then the same asymptotic formula should hold for the right-hand side of (3) as well. In this way the ruin probability Ψ(𝑥,𝑛) has the same asymptotic behavior as that of the tail probability ℙ(𝑆^𝑌_𝑛>𝑥) as 𝑥 tends to infinity.

There has been a vast amount of literature studying the asymptotic behavior of the tail probability of the randomly weighted sum 𝑆^𝑌_𝑛. Many works have considered the heavy-tailed case; that is, the distribution 𝐹 of 𝑋 belongs to some classes of heavy-tailed distributions, even under some dependence structures. For example, one can refer to Tang and Tsitsiashvili [2, 3], Wang and Tang [4], Zhang et al. [5], Shen et al. [6], Chen and Yuen [7], Gao and Wang [8], and Yi et al. [9] among others for some details in this direction, where the distribution 𝐹 is heavily heavy tailed; as for some lightly heavy-tailed distribution 𝐹, some related results were obtained by Tang and Tsitsiashvili [3, 10], Chen and Su [11], Hashorva et al. [12], Yang et al. [13], Yang and Hashorva [14], and Yang and Wang [15] among others. We pointed out that Tang and Tsitsiashvili [3] achieved some interesting results on the asymptotics for the tail probability ℙ(𝑆^𝑌_𝑛>𝑥) in some cases where 𝐹 belongs to the intersection between the subexponential distribution class and the rapidly varying distribution class.

In this paper, we aim to consider the light-tailed case, more exactly, to investigate the asymptotic behavior of the tail probability of the randomly weighted sums with increments with convolution-equivalent distributions.

Hereafter, all the limit relationships hold for 𝑥 tending to infinity. For two positive functions 𝑎(𝑥) and 𝑏(𝑥), we write 𝑎(𝑥)∼𝑏(𝑥) if lim𝑎(𝑥)/𝑏(𝑥)=1; write 𝑎(𝑥)=𝑜(𝑏(𝑥)) if lim𝑎(𝑥)/𝑏(𝑥)=0; and write 𝑎(𝑥)=𝑂(𝑏(𝑥)) if limsup𝑎(𝑥)/𝑏(𝑥)<∞.

Firstly we introduce some definitions on some classes of convolution-equivalent distributions. A distribution 𝑉 on [0,∞) belongs to the class of convolution-equivalent distributions, denoted by 𝒮(𝛾), 𝛾≥0, if for any 𝑦∈ℝ, lim𝑉𝑥−𝑦𝑉(𝑥)=𝑒^𝛾𝑦,lim𝑉^∗2(𝑥)𝑉(𝑥)=2^∞₀𝑒^𝛾𝑢𝑉(𝑑𝑢)<∞, (4)(5) where 𝑉^∗2 denotes the convolution of 𝑉 with itself. More generally, a distribution 𝑉 on ℝ belongs to the class 𝒮(𝛾), 𝛾≥0, if and only if its right-hand distribution 𝑉⁺(𝑥)=𝑉(𝑥)𝟏_{𝑥≥0} belongs to this class; see Corollary 2.1 of Pakes [16]. The class 𝒮∶=𝒮(0) is called the class of subexponential distributions. A distribution 𝑉 on ℝ belongs to the class ℒ(𝛾), 𝛾≥0 if only relation (4) holds. In the case 𝛾=0, we say that ℒ∶=ℒ(0) is the class of long-tailed distributions. Similarly, a positive function 𝑓(⋅) is said to be long tailed if lim𝑓(𝑥−𝑦)/𝑓(𝑥)=1 for any 𝑦∈ℝ. Clearly, if a distribution 𝑉∈ℒ, then its tail probability 𝑉(𝑥) is long tailed. Closely related is the class 𝒜, which was introduced by Konstantinides et al. [17]. A distribution 𝑉 on ℝ belongs to the class 𝒜 if 𝑉 is subexponential, and, for some 𝑦>1, limsup𝑉𝑥𝑦𝑉(𝑥)<1. (6) Clearly, all distributions in the classes 𝒜, 𝒮, and ℒ are heavy tailed. A distribution 𝑉 of r.v. 𝜉 is said to be heavy tailed if 𝔼𝑒^𝑠𝜉=∞ for any 𝑠>0; otherwise it is said to be light tailed.

For each 𝑛≥1, denote the distribution of 𝑋_𝑛∏^𝑛_𝑗=1𝑌_𝑗 by 𝐻_𝑛, by convention, 𝐻=𝐻₁. Now we state our main result as follows.

Theorem 1. If 𝐹∈𝒮(𝛾) for some 𝛾>0, 𝑥_𝐺=∞, and, for all 𝑢>0, 𝐺(𝑢𝑥)=𝑜𝐻(𝑥), (7) then, for each 𝑛≥1, ℙ𝑆^𝑌_𝑛>𝑥∼𝐻_𝑛(𝑥). (8)

Remark 2. We remark that Tang [18] considered a similar result for 0<𝑥_𝐺<∞, whereas Theorem 1 deals with the case with 𝑥_𝐺=∞ for a complement.

Remark 3. In Theorem 1, relation (7) is a mild condition. According to Corollary 1.1 of Tang [19], relation (7) can be further implied by either(a)𝐺(𝑣𝑥)=𝑜(𝐺(𝑥)) for some 𝑣>1 or(b)𝐺(𝑣𝑥)=𝑜(𝐹(𝑥)) for some 𝑣>0.

2. Proof of the Main Result

We start this section by a series of lemmas. The first two lemmas are due to Lemma 3.2 and Theorem 2.1 of Tang [20].

Lemma 4. For two distributions 𝐺 and 𝐻 with 𝐺(𝑥)>0 and 𝐻(𝑥)>0 for all 𝑥≥0, relation (7) holds for each 𝑢>0, if and only if there is a nonnegative function 𝑎(⋅) such that 𝑎(𝑥)𝑎(𝑥)↗∞,𝑥↘0,𝐺(𝑎(𝑥))=𝑜𝐻(𝑥). (9)

Lemma 5. Consider the product 𝑋𝑌. The distribution 𝐻 of the product belongs to the class 𝒜 if and only if 𝐹∈𝒜 and relation (7) holds for all 𝑢>0.

Tang [19] obtained an interesting result to show that a light-tailed random variable can be transferred into a heavy-tailed one through multiplier.

Lemma 6. Consider the product 𝑋𝑌 with 𝐹∈𝒮(𝛾) for some 𝛾>0 and 𝑥_𝐺=∞. If relation (7) holds for all 𝑢>0, then 𝐻∈𝒮.

The last lemma can be found in, for example, Theorem 3.14 of Foss et al. [21].

Lemma 7. Let a reference distribution 𝑉 on ℝ belong to the class 𝒮. Assume that distributions 𝑉₁,…,𝑉_𝑛 on ℝ satisfy that, for each 𝑖=1,…,𝑛, the function 𝑉+𝑉_𝑖 is long tailed and 𝑉_𝑖(𝑥)=𝑂(𝑉(𝑥)). Then, it holds that 𝑉₁∗⋯∗𝑉_𝑛(𝑥)=𝑉₁(𝑥)+⋯+𝑉_𝑛(𝑥)+𝑜𝑉(𝑥). (10)

Proof of Theorem 1. Now we begin to prove the main result of Theorem 1.
For each 𝑛≥1, write 𝑇_𝑛∶=^𝑛_𝑖=1𝑋_𝑖^𝑛_𝑗=𝑖𝑌=_𝑗^𝑑𝑆^𝑌_𝑛, (11) where =^𝑑 stands for equality in distribution. Since 𝐹∈𝒮(𝛾)⊂ℒ(𝛾), 𝛾>0, its tail distribution 𝐹 is rapidly varying in the sense that lim𝐹𝑥𝑦𝐹(𝑥)=0,∀𝑦>1. (12) By Lemma 6, we get that 𝐻∈𝒮. Further, by Lemma 4, there exists a nonnegative function 𝑎(⋅) such that (9) holds. Thus, by (9) and (12), for any 𝑦>1, limsup𝐻⁺𝑥𝑦𝐻⁺(𝑥)∫=limsup^0𝑎(𝑥)+∫^∞_𝑎(𝑥)𝐹𝑥𝑦/𝑢𝐺(d𝑢)𝐻⁺(𝑥)∫≤limsup^0𝑎(𝑥)𝐹𝑥𝑦/𝑢𝐺(d𝑢)∫₀^𝑎(𝑥)𝐹(𝑥/𝑢)𝐺(d𝑢)+limsup𝐺(𝑎(𝑥))𝐻⁺(𝑥)≤limsupsup_{0≤𝑢≤𝑎(𝑥)}𝐹𝑥𝑦/𝑢𝐹(𝑥/𝑢)=limsup𝐹𝑥𝑦𝐹(𝑥)=0<1, (13) which, together with 𝐻∈𝒮, implies that 𝐻∈𝒜.
We proceed to prove relation (8) by induction on 𝑛. Trivially, the distribution 𝐻₁=𝐻 of 𝑇₁ or 𝑆₁ belongs to the class 𝒜, and relation (8) holds for 𝑛=1. Assume that 𝐻_𝑛∈𝒜 and (8) holds for 𝑛. We aim to prove that 𝐻_𝑛+1∈𝒜 and (8) holds for 𝑛+1, which, by (11), is equivalent to ℙ𝑇_𝑛+1>𝑥∼𝐻_𝑛+1(𝑥). (14)
First of all, according to Lemma 2.17 of Foss et al. [21] and 𝐻_𝑛∈𝒜⊂ℒ, we have, that for any 𝑠>0, lim𝐻_𝑛(𝑥)𝑒^𝑠𝑥=∞, (15) which, together with 𝐹∈𝒮(𝛾),𝛾>0, implies that 𝐹(𝑥)=𝑜𝐻_𝑛(𝑥). (16) By (16) and 𝐻_𝑛∈𝒜⊂ℒ, we have that for any 𝑦∈ℝ, lim𝐻_𝑛𝑥−𝑦+𝐹𝑥−𝑦𝐻_𝑛(𝑥)+𝐹(𝑥)=lim𝐻_𝑛𝑥−𝑦𝐻_𝑛(𝑥)+𝐹𝑥−𝑦𝐻_𝑛⋅𝑥−𝑦𝐻_𝑛𝑥−𝑦𝐻_𝑛(𝑥)×1+𝐹(𝑥)𝐻_𝑛(𝑥)⁻¹=1, (17) which shows that the function 𝐻_𝑛(𝑥)+𝐹(𝑥) is long tailed. Since 𝑇_𝑛 and 𝑋_𝑛+1 are independent of each other, thus, by 𝐻_𝑛∈𝒜⊂𝒮, we can apply Lemma 7 to derive from the induction assumption and (16) that ℙ𝑇_𝑛+𝑋_𝑛+1>𝑥=𝐻_𝑛(𝑥)+𝐹(𝑥)+𝑜𝐻_𝑛(𝑥=((1)1+𝑜))𝐻_𝑛(𝑥). (18) For the above-mentioned nonnegative function 𝑎(⋅), from (9) and (18), we obtain that ℙ𝑇_𝑛+1𝑇>𝑥=ℙ_𝑛+𝑋_𝑛+1𝑌_𝑛+1=>𝑥_0𝑎(𝑥)+^∞_𝑎(𝑥)ℙ𝑇_𝑛+𝑋_𝑛+1>𝑥𝑢𝐺()=((1d𝑢1+𝑜))₀^𝑎(𝑥)𝐻_𝑛𝑥𝑢𝐺(d𝑢)+𝑜𝐻(𝑥)=(1+𝑜(1))𝐻_𝑛+1(𝑥), (19) where the last step used the fact that 𝐻(𝑥)=𝑂(𝐻_𝑛+1(𝑥)), because, for any 𝑥≥0, 𝐻_𝑛+1(𝑥)≥ℙ𝑋_𝑛+1^𝑛+1_𝑗=1𝑌_𝑗>𝑥,𝑌₁>1,…,𝑌_𝑛≥>1𝐻(𝑥)𝐺(1)^𝑛, (20) and 𝐺(1)>0 by 𝑥_𝐺=∞. Relation (19) means that (14) holds.
Finally, by (9) and (20), we have that 𝐺(𝑎(𝑥))=𝑜𝐻_𝑛+1(𝑥), (21) from which and 𝐻_𝑛∈𝒜, Lemma 5 gives that 𝐻_𝑛+1∈𝒜.
This completes the proof of Theorem 1.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

The authors are most grateful to the two referees and the editor for their very thorough reading of the paper and valuable suggestions to improve the presentation of this paper. This research is supported by the National Natural Science Foundation of China (nos. 11001052 and 71171046), China Postdoctoral Science Foundation (no. 2012M520964), the Natural Science Foundation of Jiangsu Province of China (no. BK20131339), Qing Lan Project, and Project of Construction for Superior Subjects of Statistics & Audit Science and Technology of Jiangsu Higher Education Institutions.