Research Article | Open Access

Kaiyong Wang, Yang Chen, Zhongquan Tan, "Asymptotics for the Solutions to Defective Renewal Equations", *Abstract and Applied Analysis*, vol. 2014, Article ID 732735, 5 pages, 2014. https://doi.org/10.1155/2014/732735

# Asymptotics for the Solutions to Defective Renewal Equations

**Academic Editor:**Yiming Ding

#### Abstract

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.

#### 1. Introduction

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. [2], Embrechts and Goldie [3], and Cline [4]. Asmussen [5] and Asmussen et al. [6] considered the case that is a subexponential density. Yin and Zhao [7] obtained the asymptotics of for the monotone function . For the above case, K. Wang and Y. Wang [8] gave the local asymptotics of . Cui et al. [9] considered a new case that where is a positive constant. In Corollary 5.1 and Theorem 5.2 of Cui et al. [9], 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 [10] and Chover et al. [11, 12] for distributions on and by Pakes [13] 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 [3] and Theorems 1.1 and 1.2 of Yu et al. [14]. The class is first introduced by Klüppelberg [15] and detailedly studied in Klüppelberg and Villasenor [16], Shimura and Watanabe [17], Watanabe and Yamamura [18], Lin and Wang [19], Yang and Wang [20], and Wang et al. [21], among others. This paper will consider the case that , . As noted by Wang et al. [21], for each , and the class is nonempty.

We first present the main result for the heavy-tailed case.

Theorem 1. *For the renewal equation (1), assume that (4) holds. If and , then
*

*Remark 2. *If then by Theorem 1 of Cline [4]. Hence, . The result of Theorem 1 implies that
which is Corollary 5.1 (ii) of Cui et al. [9].

In the following, we give the result for the light-tailed case.

Theorem 3. *For the renewal equation (1), assume that (4) holds. If for some satisfying and , then
**
where
*

*Remark 4. *If for some , by Theorem 1 of Cline [4], then and . Therefore, . Then we can obtain from Theorem 3 that
which is Theorem 5.2 (ii) of Cui et al. [9].

#### 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 [22], 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. [23], it holds that

When and since there exists such that
by Theorem 1.2 of Yu et al. [14], it holds that
On the other hand, since , by Fatou’s lemma and Lemma 5.4 of Pakes [13], 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 [22], 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
Hence,

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. [9], 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.

#### Acknowledgments

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.

#### References

- S. Asmussen,
*Applied Probability and Queues*, vol. 51 of*Applications of Mathematics*, Springer, New York, NY, USA, 2nd edition, 2003. View at: MathSciNet - P. Embrechts, C. M. Goldie, and N. Veraverbeke, “Subexponentiality and infinite divisibility,”
*Zeitschrift für Wahrscheinlichkeitstheorie und Verwandte Gebiete*, vol. 49, no. 3, pp. 335–347, 1979. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet - P. Embrechts and C. M. Goldie, “On convolution tails,”
*Stochastic Processes and Their Applications*, vol. 13, no. 3, pp. 263–278, 1982. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet - D. B. H. Cline, “Convolutions of distributions with exponential and subexponential tails,”
*Australian Mathematical Society Journal A: Pure Mathematics and Statistics*, vol. 43, no. 3, pp. 347–365, 1987. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet - S. Asmussen, “A probabilistic look at the Wiener-Hopf equation,”
*SIAM Review*, vol. 40, no. 2, pp. 189–201, 1998. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet - S. Asmussen, S. Foss, and D. Korshunov, “Asymptotics for sums of random variables with local subexponential behaviour,”
*Journal of Theoretical Probability*, vol. 16, no. 2, pp. 489–518, 2003. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet - C. Yin and J. Zhao, “Nonexponential asymptotics for the solutions of renewal equations, with applications,”
*Journal of Applied Probability*, vol. 43, no. 3, pp. 815–824, 2006. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet - K. Wang and Y. Wang, “Equivalent conditions of local asymptotics for the solutions of defective renewal equations, with applications,”
*Acta Mathematicae Applicatae Sinica (English Series)*, vol. 26, no. 3, pp. 503–512, 2010. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet - Z. Cui, Y. Wang, and K. Wang, “Asymptotics for the moments of the overshoot and undershoot of a random walk,”
*Advances in Applied Probability*, vol. 41, no. 2, pp. 469–494, 2009. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet - V. P. Chistyakov, “A theorem on sums of independent positive random variablesand its applications to branching process,”
*Theory of Probability and Its Applications*, vol. 9, no. 4, pp. 640–648, 1964. View at: Publisher Site | Google Scholar - J. Chover, P. Ney, and S. Wainger, “Degeneracy properties of subcritical branching processes,”
*Annals of Probability*, vol. 1, pp. 663–673, 1973. View at: Google Scholar | Zentralblatt MATH | MathSciNet - J. Chover, P. Ney, and S. Wainger, “Functions of probability measures,”
*Journal d'Analyse Mathématique*, vol. 26, pp. 255–302, 1973. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet - A. G. Pakes, “Convolution equivalence and infinite divisibility,”
*Journal of Applied Probability*, vol. 41, no. 2, pp. 407–424, 2004. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet - C. Yu, Y. Wang, and Z. Cui, “Lower limits and upper limits for tails of random sums supported on $\mathbb{R}$,”
*Statistics and Probability Letters*, vol. 80, no. 13-14, pp. 1111–1120, 2010. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet - C. Klüppelberg, “Asymptotic ruin probabilities and hazard rates,”
*Mathematics of Operations Research*, vol. 60, pp. 567–576, 1990. View at: Google Scholar - C. Klüppelberg and J. A. Villasenor, “The full solution of the convolution closure problem for convolution-equivalent distributions,”
*Journal of Mathematical Analysis and Applications*, vol. 160, no. 1, pp. 79–92, 1991. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet - T. Shimura and T. Watanabe, “Infinite divisibility and generalized subexponentiality,”
*Bernouli*, vol. 11, no. 3, pp. 445–469, 2005. View at: Publisher Site | Google Scholar - T. Watanabe and K. Yamamura, “Ratio of the tail of an infinitely divisible distribution on the line to that of its Lévy measure,”
*Electronic Journal of Probability*, vol. 15, no. 2, pp. 44–74, 2010. View at: Google Scholar - J. Lin and Y. Wang, “New examples of heavy-tailed O-subexponential distributions and related closure properties,”
*Statistics and Probability Letters*, vol. 82, no. 3, pp. 427–432, 2012. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet - Y. Yang and K. Wang, “Estimates for the tail probability of the supremum of a random walk with independent increments,”
*Chinese Annals of Mathematics B*, vol. 32, no. 6, pp. 847–856, 2011. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet - K. Wang, Y. Yang, and C. Yu, “Estimates for the overshoot of a random walk with negative drift and non-convolution equivalent increments,”
*Statistics and Probability Letters*, vol. 83, no. 6, pp. 1504–1512, 2013. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet - C. Yu and Y. Wang, “Tail behavior of supremum of a random walk when Cramér's condition fails,”
*Frontiers of Mathematics in China*, vol. 9, no. 2, pp. 431–453, 2014. View at: Publisher Site | Google Scholar | MathSciNet - D. Denisov, S. Foss, and D. Korshunov, “On lower limits and equivalences for distribution tails of randomly stopped sums,”
*Bernoulli*, vol. 14, no. 2, pp. 391–404, 2008. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet

#### Copyright

Copyright © 2014 Kaiyong Wang et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.