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Asymptotics for the moments of the overshoot and undershoot of a random walk

Part of: Special processes

Published online by Cambridge University Press:  01 July 2016

Zhaolei Cui ,
Yuebao Wang  and
Kaiyong Wang
Zhaolei Cui*
Affiliation:
Soochow University
Yuebao Wang*
Affiliation:
Soochow University
Kaiyong Wang*
Affiliation:
Suzhou University of Science and Technology
*
∗ Postal address: Department of Mathematics, Soochow University, Suzhou, 215006, P. R. China.
∗ Postal address: Department of Mathematics, Soochow University, Suzhou, 215006, P. R. China.
∗∗∗ Postal address: Department of Information and Computational Science, School of Mathematics and Physics, Suzhou University of Science and Technology, Suzhou, 215009, P. R. China.
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Abstract

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In this paper we obtain some equivalent conditions and sufficient conditions for the local and nonlocal asymptotics of the φ-moments of the overshoot and undershoot of a random walk, where φ is a nonnegative, long-tailed function. By the strong Markov property, it can be shown that the moments of the overshoot and undershoot and the moments of the first ascending ladder height of a random walk satisfy some renewal equations. Therefore, in this paper we first investigate the local and nonlocal asymptotics for the moments of the first ascending ladder height of a random walk, and then give some equivalent conditions and sufficient conditions for the asymptotics of the solutions to some renewal equations. Using the above results, the main results of this paper are obtained.

Keywords

Local and nonlocal asymptoticsmomentfirst ascending ladder heightrenewal equationovershootundershoot

MSC classification

Primary: 60K05: Renewal theory
Secondary: 60K10: Applications (reliability, demand theory, etc.) 60K30: Applications (congestion, allocation, storage, traffic, etc.)
Type
General Applied Probability
Information
Advances in Applied Probability , Volume 41 , Issue 2 , June 2009 , pp. 469 - 494
Copyright
Copyright © Applied Probability Trust 2009 

References

Asmussen, S. (1998). A probabilistic look at the Wiener–Hopf equation. SIAM Rev. 40, 189–201.CrossRefGoogle Scholar
Asmussen, S. (2003). Applied Probability and Queues (Appl. Math. 51), 2nd edn. Springer, New York.Google Scholar
Asmussen, S., Foss, S. and Korshunov, D. (2003). Asymptotics for sums of random variables with local subexponential behavior. J. Theoret. Prob. 16, 489–518.CrossRefGoogle Scholar
Asmussen, S. et al. (2002). A local limit theorem for random walk maxima with heavy tails. Statist. Prob. Lett. 56, 399–404.CrossRefGoogle Scholar
Bertoin, J. and Doney, R. A. (1996). Some asymptotic results for transient random walks. Adv. Appl. Prob. 28, 207–226.CrossRefGoogle Scholar
Borovkov, A. A. (1976). Stochastic Processes in Queueing Theory (Appl. Math. 4). Springer, Berlin.CrossRefGoogle Scholar
Borovkov, A. A. and Foss, S. (2000). Estimates for overshooting an arbitrary boundary by a random walk and their applications. Theory Prob. Appl. 44, 231–253.CrossRefGoogle Scholar
Cai, J. and Garrido, J. (2002). Asymptotic forms and bounds for tails of convolutions of compound geometric distributions, with applications. In Recent Advances in Statistical Methods, ed. Chaubey, Y. P., Imperial College Press, London, pp. 114–131.CrossRefGoogle Scholar
Cai, J. and Tang, Q. (2004). On max-sum equivalence and convolution closure of heavy-tailed distributions and their applications. J. Appl. Prob. 41, 117–130.CrossRefGoogle Scholar
Cheng, Y., Tang, Q. and Yang, H. (2002). Approximations for moments of deficit at ruin with exponential and subexponential claims. Statist. Prob. Lett. 59, 367–378.CrossRefGoogle Scholar
Chen, G., Wang, Y. and Cheng, F. (2009). The uniform local asymptotics of the overshoot of a random walk with heavy-tailed increments. To appear in Stoch. Models.CrossRefGoogle Scholar
Chistyakov, V. P. (1964). A theorem on sums of independent positive random variables and its applications to branching process. Theory Prob. Appl. 9, 640–648.CrossRefGoogle Scholar
Chover, J., Ney, P. and Wainger, S. (1973a). Degeneracy properties of subcritical branching processes. Ann. Prob. 1, 663–673.CrossRefGoogle Scholar
Chover, J., Ney, P. and Wainger, S. (1973b). Functions of probability measures. J. Analyse Math. 26, 255–302.CrossRefGoogle Scholar
Cline, D. B. H. (1987). Convolutions of distributions with exponential and subexponential tails. J. Austral. Math. Soc. Ser. A 43, 347–365.CrossRefGoogle Scholar
Cui, Z. and Wang, Y. (2007). A simple proof of approximations for moments of deficit at ruin. J. Suzhou Univ. 23, 23–25 (in Chinese).Google Scholar
Denisov, D. and Shneer, V. (2007). Local asymptotics of the cycle maximum of a heavy-tailed random walk. Adv. Appl. Prob. 39, 221–244.CrossRefGoogle Scholar
Denisov, D., Foss, S. and Korshunov, D. (2004). Tail asymptotics for the supremum of a random walk when the mean is not finite. Queueing Systems 46, 15–33.CrossRefGoogle Scholar
Doney, R. A. and Kyprianou, A. E. (2006). Overshoots and undershoots of Lévy processes. Ann. Appl. Prob. 16, 91–106.CrossRefGoogle Scholar
Embrechts, P. and Goldie, C. M. (1982). On convolution tails. Stoch. Process. Appl. 13, 263–278.CrossRefGoogle Scholar
Embrechts, P., Goldie, C. M. and Veraverbeke, N. (1979). Subexponentiality and infinite divisibility. Z. Wahrscheinlichkeitsth. 49, 335–347.CrossRefGoogle Scholar
Embrechts, P., Klüppelberg, C. and Mikosch, T. (1997). Modelling Extremal Events (Appl. Math. 33). Springer, Berlin.CrossRefGoogle Scholar
Foss, S. and Korshunov, D. (2007). Lower limits and equivalences for convolution tails. Ann. Prob. 35, 366–383.CrossRefGoogle Scholar
Foss, S. and Zachary, S. (2003). The maximum on a random time interval of a random walk with long-tailed increments and negative drift. Ann. Appl. Prob. 13, 37–53.CrossRefGoogle Scholar
Gao, Q. and Wang, Y. (2009). Ruin probability and local ruin probability in the random multi-delayed renewal risk model. Statist. Prob. Lett. 79, 588–596.CrossRefGoogle Scholar
Janson, S. (1986). Moments for first-passage and last-exit times, the minimum, and related quantities for random walks with positive drift. Adv. Appl. Prob. 18, 865–879.CrossRefGoogle Scholar
Klüppelberg, C. (1988). Subexponential distributions and integrated tails. J. Appl. Prob. 25, 132–141.CrossRefGoogle Scholar
Klüppelberg, C. (1989). Subexponential distributions and characterizations of related classes. Prob. Theory Relat. Fields 82, 259–269.CrossRefGoogle Scholar
Klüppelberg, C., Kyprianou, A. E. and Maller, R. A. (2004). Ruin probabilities and overshoots for general Lévy insurance risk processes. Ann. Appl. Prob. 14, 1766–1801.CrossRefGoogle Scholar
Pakes, A. G. (2004). Convolution equivalence and infinite divisibility. J. Appl. Prob. 41, 407–424.CrossRefGoogle Scholar
Park, H. S. and Maller, R. (2008). Moment and MGF convergence of overshoots and undershoots for Lévy insurance risk processes. Adv. Appl. Prob. 40, 716–733.CrossRefGoogle Scholar
Tang, Q. (2007). The overshoot of a random walk with negative drift. Statist. Prob. Lett. 77, 158–165.CrossRefGoogle Scholar
Veraverbeke, N. (1977). Asymptotic behaviour of Wiener–Hopf factors of a random walk. Stoch. Process. Appl. 5, 27–37.CrossRefGoogle Scholar
Wang, K., Wang, Y. and Yin, C. (2008). Equivalent conditions of local asymptotics for the overshoot of a random walk with heavy-tailed increments. Submitted.Google Scholar
Wang, Y. and Wang, K. (2006). Asymptotics of the density of the supremum of a random walk with heavy-tailed increments. J. Appl. Prob. 43, 874–879.CrossRefGoogle Scholar
Wang, Y. and Wang, K. (2009). Equivalent conditions of asymptotics for the density of the supremum of a random walk in the intermediate case. J. Theoret. Prob. 22, 281–293.CrossRefGoogle Scholar
Wang, Y., Cheng, D. and Wang, K. (2005). The closure of a local subexponential distribution class under convolution roots with applications to the compound Poisson process. J. Appl. Prob. 42, 1194–1203.CrossRefGoogle Scholar
Wang, Y., Yang, Y., Wang, K. and Cheng, D. (2007). Some new equivalent conditions on asymptotics and local asymptotics for random sums and their applications. Insurance Math. Econom. 40, 256–266.CrossRefGoogle Scholar
Yin, C. and Zhao, J. (2006). Nonexponential asymptotics for the solutions of renewal equations, with applications. J. Appl. Prob. 43, 815–824.CrossRefGoogle Scholar
Yu, C., Wang, Y. and Cui, Z. (2009). Lower limits and upper limits for tails of random sums supported on R . Submitted.Google Scholar