EQUIVALENT CONDITIONS OF LOCAL ASYMPTOTICS FOR THE OVERSHOOT OF A RANDOM WALK WITH HEAVY-TAILED INCREMENTS

doi:10.1016/S0252-9602(11)60213-0

Abstract

Abstract:

This article gives the equivalent conditions of the local asymptotics for the overshoot of a random walk with heavy-tailed increments,
from which we find that the above asymptotics are different from the local asymptotics for the supremum of the random walk. To do
this, the article first extends and improves some existing results about the solutions of renewal equations.

Key words: Equivalent conditions, asymptotics, random walk, renewal equation

CLC Number:

60K05

WANG Kai-Yong, WANG Yue-Bao, YIN Chuan-Cun. EQUIVALENT CONDITIONS OF LOCAL ASYMPTOTICS FOR THE OVERSHOOT OF A RANDOM WALK WITH HEAVY-TAILED INCREMENTS[J].Acta mathematica scientia,Series B, 2011, 31(1): 109-116.

Trendmd

References

[1] Asmussen S. A probabilistic look at the Wiener-Hopf equation. SIAM Rev, 1998, 40: 189--201

[2] Asmussen S. Applied Probability and Queues. 2nd ed. New York: Springer, 2003

[3] Asmussen S, Foss S, Korshunov D. Asymptotics for sums of random variables with local subexponential behavior. J Theor Prob,
2003, 16: 489--518

[4] Asmussen S, Kalashnikov V, Konstantinides D, et al. A local limit theorem for random walk maxima with heavy tails. Statist Prob Lett, 2002, 56: 399--404

[5] Cai J, Garrido J. Asymptotic forms and tails of convolutions of compound geometric distributions, with applications//Chaubey Y P.
Recent Advances in Statistical Methods. London: Imperial College Press, 2002

[6] Cai J, Tang Q. On max-sum equivalence and convolution closure of heavy-tailed distributions and their applications. J Appl Prob, 2004, 41: 117--130

[7] Chen G, Wang Y, Cheng F. The uniform local asymptotics of the overshoot of a random walk with heavy-tailed increments. Stochastic Models, 2009, 25: 508--521

[8] Cline D B H. Convolutions of distributions with exponential and subexponential tails. J Aust Math Soc (Series A), 1987, 43: 347--365

[9] Cui Z, Wang Y, Wang K. Asymptotics for moments of the overshoot and undershoot of a random walk. Adv Appl Prob, 2009, 41: 469--494

[10] Embrechts P, Goldie C M, Veraverbeke N. Subexponentiality and infinite divisibility. Z Wahrscheinlichkeitstheorie und verw Gebiete, 1979, 49: 335--347

[11] Embrechts P, Goldie C M. On convolution tails. Stoc Proc Appl, 1982, 13: 263--270

[12] Embrechts P, Kl\"{u}ppelberg C, Mikosch T. Modelling Extremal Events for Insurance and Finance. Berlin: Springer, 1997

[13] Feller W. An introduction to probability theory and its applications. Volume 2 (second edition). New York: Wiley, 1971

[14] Foss S, Zachary S. The maximum on a random time interval of a random walk with long-tailed increments and negative drift. Ann Appl Prob, 2003, 13: 37--53

[15] Gao Q, Wang Y. Ruin probability and local ruin probability in the random multi-delayed renewal risk model. Statist Prob Lett, 2009, 79: 588--596

[16] Kl\"{u}ppelberg C. Subexponential distributions and characterization of related classes. Probab Theory and Related Fields, 1989, 82: 259--269.

[17] Wang Y, Cheng D, Wang K. The closure of a local subexponential distribution class under convolution roots with applications to the
compound Poisson process. J Appl Prob, 2005, 42: 1194--1203

[18] Wang Y, Wang K. Asymptotics for the density of the supremum of a random walk with heavy-tailed increments. J Appl Prob, 2006, 43: 874--879

[19] Wang Y, Yang Y, Wang K, et al. Some new equivalent conditions on asymptotics and local asymptotics for random sums and their applications. Insurance Mathematics and Economics, 2007, 40: 256--266

[20] Willmot G, Cai J, Lin X S. Lundberg inequalities for renewal equations. Adv Appl Prob, 2001, 33: 674--689

[21] Yin C, Zhao J. Nonexponential asymptotics for the solutions of renewal equations, with applications. J Appl Prob, 2006, 43: 815--824

[22] Yin C, Zhao X, Hu F. Ladder height and supremum of a random walk with applications in risk theory. Acta Mathematica Scientia, 2009, 29A: 38--47