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On the Maximum Exceedance of a Sequence of Random Variables Over a Renewal Threshold

Published online by Cambridge University Press:  14 July 2016

Xuemiao Ha*
Affiliation:
The University of Iowa
Qihe Tang*
Affiliation:
The University of Iowa
Li Wei*
Affiliation:
Renmin University of China
*
Postal address: Department of Statistics and Actuarial Science, The University of Iowa, 241 Schaeffer Hall, Iowa City, IA 52242, USA.
Postal address: Department of Statistics and Actuarial Science, The University of Iowa, 241 Schaeffer Hall, Iowa City, IA 52242, USA.
∗∗∗∗Postal address: School of Finance, Renmin University of China, Beijing, 100872, P. R. China. Email address: [email protected]
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Abstract

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In this paper we study the tail behavior of the maximum exceedance of a sequence of independent and identically distributed random variables over a random walk. For both light-tailed and heavy-tailed cases, we derive a precise asymptotic formula, which extends and unifies some existing results in the recent literature of applied probability.

Type
Research Article
Copyright
Copyright © Applied Probability Trust 2009 

References

Araman, V. F. and Glynn, P. W. (2006). Tail asymptotics for the maximum of perturbed random walk. Ann. Appl. Prob. 16, 14111431.Google Scholar
Asmussen, S., Schmidli, H. and Schmidt, V. (1999). Tail probabilities for non-standard risk and queueing processes with subexponential Jumps. Adv. Appl. Prob. 31, 422447.Google Scholar
Bingham, N. H., Goldie, C. M. and Teugels, J. L. (1987). Regular Variation (Encyclopaedia Math Appl. 27). Cambridge University Press.CrossRefGoogle Scholar
Pakes, A. G. (2004). Convolution equivalence and infinite divisibility. J. Appl. Prob. 41, 407424.CrossRefGoogle Scholar
Palmowski, Z. and Zwart, B. (2007). Tail asymptotics of the supremum of a regenerative process. J. Appl. Prob. 44, 349365.CrossRefGoogle Scholar
Robert, C. Y. (2005). Asymptotic probabilities of an exceedance over renewal thresholds with an application to risk theory. J. Appl. Prob. 42, 153162.CrossRefGoogle Scholar
Rogozin, B. A. and Sgibnev, M. S. (1999). Banach algebras of measures on the line with given asymptotics of distributions at infinity. Siberian Math. J. 40, 565576.CrossRefGoogle Scholar
Su, C. and Tang, Q. (2003). Characterizations on heavy-tailed distributions by means of hazard rate. Acta Math. Appl. Sin. Engl. Ser. 19, 135142.Google Scholar
Tang, Q. (2007). The overshoot of a random walk with negative drift. Statist. Prob. Lett. 77, 158165.Google Scholar
Veraverbeke, N. (1977). Asymptotic behaviour of Wiener–Hopf factors of a random walk. Stoch. Process. Appl. 5, 2737.Google Scholar