Asymptotics for solutions of a defective renewal
equation with applications

doi:10.1007/s11464-008-0024-y

Front. Math. China

2008, Vol. 3

Issue (3) : 443-459 https://doi.org/10.1007/s11464-008-0024-y

Asymptotics for solutions of a defective renewal equation with applications

YIN Chuancun, ZHAO Xianghua

School of Mathematical Sciences, Qufu Normal University;

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Abstract In this paper, we derive non-exponential asymptotic forms for solutions of defective renewal equations. These include as special cases asymptotics for compound geometric distribution and the convolution of a compound geometric distribution with a distribution function. As applications of these results, we study the Gerber-Shiu discounted penalty function in the classical risk model and the reliability of a two-unit cold standby system in reliability theory.

Issue Date: 05 September 2008

Cite this article:

YIN Chuancun,ZHAO Xianghua. Asymptotics for solutions of a defective renewal equation with applications[J]. Front. Math. China, 2008, 3(3): 443-459.

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https://academic.hep.com.cn/fmc/EN/10.1007/s11464-008-0024-y
https://academic.hep.com.cn/fmc/EN/Y2008/V3/I3/443

1	Asmussen S . Aprobabilistic look at the Wiener-Hopf equation. SIAM Review, 1998, 40: 189–201. doi:10.1137/S0036144596303534
2	Asmussen S, Foss S, Korshunov D . Asymptotics for sums of random variables with local subexponentialbehaviour. J Theor Probab, 2003, 16: 489–518. doi:10.1023/A:1023535030388
3	Cai J, Garrido J . Asymptotic forms and tailsof convolutions of compound geometric distributions, with applications. In: Chaubed Y P, ed. Recent Advances in Statistical Methods. London: Imperial College Press, 1999, 114–131
4	Cai J, Tang Q H . On max-sum equivalence andconvolution closure of heavy-tailed distributions and their applications. J Appl Probab, 2004, 41: 117–130. doi:10.1239/jap/1077134672
5	Cheng K, Wang W Y . Approximations of a two-unitcold standby system's reliability behavior. Microelectronics and Reliability, 1996, 36(5): 627–630. doi:10.1016/0026‐2714(95)00170‐0
6	Cheng Y, Tang Q H . Moments of the surplus beforeruin and the deficit at ruin in the Erlang(2) risk process. North American Actuar J, 2003, 7: 1–12
7	Cline D B H . Convolutions of distributions with exponential and subexponentialtails. J Austral Math Soc (Ser A), 1987, 43: 347–365
8	Embrechts P, Goldie C M . On convolution tails. Stoch Proc Appl, 1982, 13: 263–278. doi:10.1016/0304‐4149(82)90013‐8
9	Embrechts P, Goldie C M, Veraverbeke N . Subexponentiality and infinite divisibility. Z Wahrscheinlichkeitstheorie und verw Gebiete, 1979, 49: 335–347. doi:10.1007/BF00535504
10	Embrechts P, Klüppelberg C, Mikosch T . Modelling Extremal Events for Insurance and Finance. New York: Springer, 1997
11	Embrechts P, Veraverbeke N . Estimates for the probabilityof ruin with special emphasis on the possibility of large claims. Insurance Math Econom, 1982, 1: 55–72. doi:10.1016/0167‐6687(82)90021‐X
12	Feller W . AnIntroduction to Probability Theory and Its Applications, Vol 2. 2nd ed. New York: John Wiley, 1971
13	Gerber H . AnIntroduction to Mathematical Risk Theory. Philadelphia: University of Pennsylvania, 1979
14	Gerber H U, Shiu E S W . On the time value of ruin. North American Actuar J, 1998, 2: 48–72
15	Gerber H U, Shiu E SW . The time value of ruin ina Sparre Andersen model. North AmericanActuar J, 2005, 9: 49–69
16	Klüppelberg C . Onsubexponential distributions and integrated tails. J Appl Probab, 1988, 5: 132–141. doi:10.2307/3214240
17	Klüppelberg C . Subexponentialdistributions and characterizations of related classes. Probab Theory Relat Fields, 1989, 82: 259–269. doi:10.1007/BF00354763
18	Li S M, Garrido J . On ruin for the Erlang(n) risk process. Insurance Math Econom, 2004, 34: 391–408. doi:10.1016/j.insmatheco.2004.01.002
19	Rolski T, Schmidli H, Schmidt V, et al.. Stochastic Processes for Insurance and Finance. New York: Wiley, 1999
20	Ross S . Boundson the delay distribution in GI/G/1 queues. J Appl Probab, 1974, 11: 417–421. doi:10.2307/3212767
21	Ross S . StochasticProcesses. 2nd ed. New York: Wiley, 1996
22	Willmot G, Cai J, Lin X S . Lundberg inequalities for renewal equations. Adv Appl Probab, 2001, 33: 674–689. doi:10.1239/aap/1005091359
23	Yin C C . A local limit theorem for the probability of ruin. Sci Chin, Ser A, 2004, 47(5): 711–721. doi:10.1360/03ys0039
24	Yin C C, Zhao J S . Non-exponential asymptoticsfor the solutions of renewal equations with applications. J Appl Probab, 2006, 43(3): 815–824. doi:10.1239/jap/1158784948

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