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

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

数学物理学报(英文版) ›› 2011, Vol. 31 ›› Issue (1): 109-116.doi: 10.1016/S0252-9602(11)60213-0

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

王开永^1,2|王岳宝¹|尹传存³

1. Department of Mathematics, Soochow University, Suzhou 215006, China; 2. School of Mathematics and Physics, Suzhou University of Science and Technology, Suzhou 215009, China; 3. Department of Mathematics, Qufu Normal University, Qufu 273165, China

收稿日期:2009-04-23 出版日期:2011-01-20 发布日期:2011-01-20
基金资助:
This work was supported by National Science Foundation of China (10671139, 11071182, 10771119), Natural Science Foundation of Jiangsu Highter Eduction Institution of China (10KJB110010), and the research foundation of SUST.

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

WANG Kai-Yong^1,2, WANG Yue-Bao¹, YIN Chuan-Cun³

Received:2009-04-23 Online:2011-01-20 Published:2011-01-20
Supported by:
This work was supported by National Science Foundation of China (10671139, 11071182, 10771119), Natural Science Foundation of Jiangsu Highter Eduction Institution of China (10KJB110010), and the research foundation of SUST.

摘要/Abstract

摘要：

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.

关键词: Equivalent conditions, asymptotics, random walk, renewal equation

Abstract:

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

中图分类号:

60K05

王开永, 王岳宝, 尹传存. EQUIVALENT CONDITIONS OF LOCAL ASYMPTOTICS FOR THE OVERSHOOT OF A RANDOM WALK WITH HEAVY-TAILED INCREMENTS[J]. 数学物理学报(英文版), 2011, 31(1): 109-116.

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.

参考文献

[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

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

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

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