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Asymptotics for the Solutions to Defective Renewal Equations
This paper investigates the defective renewal equations under the nonconvolution equivalent distribution class. The asymptotics of the solution to the defective renewal equations have been given for the heavy-tailed and light-tailed cases, respectively.
This paper will consider the defective renewal equation where is a proper distribution on , is a known and locally bounded function on , and . The only solution to (1) is given by where (see e.g., Asmussen [1, Chapter V]) and here is the -fold convolution of with itself, , , and is the distribution degenerate at zero.
Since in most cases it is not easy to calculate (2), more attention is paid to the asymptotics of the solution . When , the asymptotics of have been investigated by many researchers, such as Embrechts et al. , Embrechts and Goldie , and Cline . Asmussen  and Asmussen et al.  considered the case that is a subexponential density. Yin and Zhao  obtained the asymptotics of for the monotone function . For the above case, K. Wang and Y. Wang  gave the local asymptotics of . Cui et al.  considered a new case that where is a positive constant. In Corollary 5.1 and Theorem 5.2 of Cui et al. , they obtained the asymptotics of under the condition that and for some with , respectively. The classes and , (the definitions of these distribution classes will be given below) are convolution equivalent distribution classes. But beyond the convolution equivalent distribution classes, there exist some other distributions. How to estimate the asymptotics of the solution for the nonconvolution equivalent distribution will be an interesting question. This paper will investigate this case. Under the conditions (4) and that may not belong to the convolution equivalent distribution class, this paper obtains the asymptotics of the solution . In order to better illuminate our motivation and results, we will introduce some notions and notation.
Without special statement, in this paper a limit is taken as . For two nonnegative functions and , we write if , write if , write if , and write if . For a proper distribution on , the tail of is . For a real number , denote by the moment generating function of .
Firstly, we will introduce some heavy-tailed and light-tailed distribution classes. Say that a random variable (r.v.) (or its corresponding distribution ) is heavy-tailed if for all , ; otherwise, say that it is light-tailed. Let be a distribution on . Say that the distribution belongs to the class for some , if for any , where, when and is a lattice distribution, and are both taken as a multiple of the lattice step. Say that the distribution belongs to the class for some , if , , and The class , is called the convolution equivalent distribution class and was introduced by  and Chover et al. [11, 12] for distributions on and by Pakes  for distributions on . Especially, we call and the subexponential distribution class and the long-tailed distribution class, denoted by and , respectively.
This paper will mainly investigate the case that the distribution may not be convolution equivalent. We will introduce another distribution class. Say that the distribution belongs to the class , if for sufficiently large and Clearly, if for some then . Therefore, for each , . If for some then , which can be obtained by Lemma 2.4 of Embrechts and Goldie  and Theorems 1.1 and 1.2 of Yu et al. . The class is first introduced by Klüppelberg  and detailedly studied in Klüppelberg and Villasenor , Shimura and Watanabe , Watanabe and Yamamura , Lin and Wang , Yang and Wang , and Wang et al. , among others. This paper will consider the case that , . As noted by Wang et al. , for each , and the class is nonempty.
We first present the main result for the heavy-tailed case.
In the following, we give the result for the light-tailed case.
2. Proofs of Theorems
Before giving the proof of Theorems 1 and 3, we first give some lemmas. The first lemma comes from Lemma 2.2 of Yu and Wang , which will need the following notation. For a distribution on and any , define
Lemma 5. Suppose that is a distribution on and belongs to the class for some . Then for any ,
Lemma 6. For the random sum (3), assume that for some . When , let ; when , let and . Then ,
Proof. We first prove (15). Let be a r.v. with a distribution . When and since is heavy-tailed and is light-tailed, by Theorem 2 of Denisov et al. , it holds that
When and since there exists such that by Theorem 1.2 of Yu et al. , it holds that On the other hand, since , by Fatou’s lemma and Lemma 5.4 of Pakes , we have This completes the proof of (15).
Now we prove (16). Since for and for , there exists such that for , Hence, by Corollary 1 of Yu and Wang , we get and (16) holds.
Proof of Theorem 1. Since , we get by Lemma 6. For any fixed positive constant , when is sufficiently large, we get
By (4) and , we get
For , since , by and Lemma 6, it holds that
For , we first estimate the asymptotics of as firstly letting and then letting . For any , it holds that Since , we have Hence, means that which, combining with (4) and Lemma 6, yields that Hence, (8) can be obtained by (22)–(25) and (30).
Proof of Theorem 3. It follows from Lemma 6 and that . Taking , when is sufficiently large, we get By (4) and , we have By (4) and Lemmas 5 and 6, it holds that For , using Lemma 6 and the way of dealing with in Theorem 5.2 of Cui et al. , we can get Hence, (10) can be obtained by (31)–(34).
Conflict of Interests
The authors declare that there is no conflict of interests regarding the publication of this paper.
The authors wish to thank the editor and the referees for their very valuable comments by which the presentation of the paper has been improved. This work is supported by the National Natural Science Foundation of China, Tian Yuan Foundation (no. 11226211), the Natural Science Foundation of Jiangsu Province (no. BK2012165), China Postdoctoral Science Foundation (no. 2012M520963), and the Research Foundation of SUST.
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