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UNIFORM ESTIMATES FOR THE TAIL PROBABILITY OF MAXIMA OVER FINITE HORIZONS WITH SUBEXPONENTIAL TAILS

Published online by Cambridge University Press:  22 January 2004

Qihe Tang
Qihe Tang
Affiliation:
Department of Quantitative Economics, University of Amsterdam, Amsterdam, The Netherlands, E-mail: q.tang@uva.nl
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Abstract

Let F be the common distribution function of the increments of a random walk {Sn, n ≥ 0} with S0 = 0 and a negative drift and let {N(t), t ≥ 0} be a general counting process, independent of {Sn, n ≥ 0}. This article investigates the tail probability, denoted by ψ(x; t), of the maximum of SN(v) over a finite horizon 0 ≤ v ≤ t. When F is strongly subexponential, some asymptotics for ψ(x; t) are derived as x → ∞. The merit is that all of the obtained asymptotics are uniform for t in a finite or infinite time interval.

Type
Research Article
Information
Probability in the Engineering and Informational Sciences , Volume 18 , Issue 1 , January 2004 , pp. 71 - 86
Copyright
© 2004 Cambridge University Press

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References

REFERENCES

Asmussen, S. (1987). Applied probability and queues. Chichester: Wiley.
Asmussen, S. (2000). Ruin probabilities. River Edge, NJ: World Scientific.CrossRef
Björk, T. & Grandell, J. (1988). Exponential inequalities for ruin probabilities in the Cox case. Scandanavian Actuarial Journal 1–2: 77–111.Google Scholar
Borovkov, A.A. & Borovkov, K.A. (2002). On probabilities of large deviations for random walks. I. Regularly varying distribution tails. Theory of Probability and Its Applications 46(2): 193–213.Google Scholar
Chistyakov, V.P. (1964). A theorem on sums of independent positive random variables and its applications to branching random processes. Theory of Probability and Its Applications 9: 710–718.Google Scholar
Denisov, D., Foss, S. & Korshunov, D. (2003). Tail asymptotics for the supremum of a random walk when the mean is not finite. Queueing Systems Theory and Applications, to appear.
Embrechts, P., Klüppelberg, C., & Mikosch, T. (1997). Modelling extremal events for insurance and finance. Berlin: Springer-Verlag.CrossRef
Embrechts, P. & Veraverbeke, N. (1982). Estimates for the probability of ruin with special emphasis on the possibility of large claims. Insurance Mathematics and Economics 1(1): 55–72.Google Scholar
Feller, W. (1971). An introduction to probability theory and its applications. Vol. 2, 2nd ed. New York: Wiley.
Foss, S. & Zachary, S. (2003). The maximum on a random time interval of a random walk with long-tailed increments and negative drift. Annals of Applied Probability 13(1): 37–53.Google Scholar
Grandell, J. (1976). Doubly stochastic Poisson processes, Lecture Notes in Mathematics Vol. 529, Berlin: Springer-Verlag.CrossRef
Kaas, R. & Tang, Q. (2003). Note on the tail behavior of random walk maxima with heavy tails and negative drift. North American Actuarial Journal 7(3): 57–61.Google Scholar
Klüppelberg, C. (1988). Subexponential distributions and integrated tails. Journal of Applied Probability 25(1): 132–141.Google Scholar
Klüppelberg, C. & Mikosch, T. (1997). Large deviations of heavy-tailed random sums with applications in insurance and finance. Journal of Applied Probability 34(2): 293–308.Google Scholar
Korolev, V.Y. (1999). On the convergence of distributions of generalized Cox processes to stable laws. Theory of Probability and Its Applications 43(4): 644–650.Google Scholar
Korolev, V.Y. (2001). Asymptotic properties of the extrema of generalized Cox processes and their application to some problems in financial mathematics. Theory of Probability and Its Applications 45(1): 136–147.Google Scholar
Korshunov, D.A. (1997). On distribution tail of the maximum of a random walk. Stochastic Processes and Their Applications 72(1): 97–103.Google Scholar
Korshunov, D.A. (2002). Large-deviation probabilities for maxima of sums of independent random variables with negative mean and subexponential distribution. Theory of Probability and Its Applications 46(2): 355–366.Google Scholar
Mikosch, T. & Nagaev, A.V. (1998). Large deviations of heavy-tailed sums with applications in insurance. Extremes 1(1): 81–110.Google Scholar
Ng, K.W., Tang, Q.H., Yan, J.A., & Yang, H. (2003). Precise large deviations for the prospective-loss process. Journal of Applied Probability 40(2): 391–400.Google Scholar
Rolski, T., Schmidli, H., Schmidt, V., & Teugels, J. (1999). Stochastic processes for insurance and finance. Chichester: Wiley.
Tang, Q., Su, C., Jiang, T., & Zhang, J.S. (2001). Large deviations for heavy-tailed random sums in compound renewal model. Statistics and Probability Letters 52(1): 91–100.Google Scholar
Veraverbeke, N. (1977). Asymptotic behaviour of Wiener–Hopf factors of a random walk. Stochastic Processes and Their Applications 5(1): 27–37.Google Scholar