Abstract
We consider the sums S n =ξ1+⋯+ξ n of independent identically distributed random variables. We do not assume that the ξ's have a finite mean. Under subexponential type conditions on distribution of the summands, we find the asymptotics of the probability P{M>x} as x→∞, provided that M=sup {S n ,n≥1} is a proper random variable. Special attention is paid to the case of tails which are regularly varying at infinity. We provide some sufficient conditions for the integrated weighted tail distribution to be subexponential. We supplement these conditions by a number of examples which cover both the infinite- and the finite-mean cases. In particular, we show that the subexponentiality of distribution F does not imply the subexponentiality of its integrated tail distribution F I.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
S. Asmussen, Applied Probability and Queues (Wiley, Chichester, 1987).
S. Asmussen, Ruin Probabilities (World Scientific, Singapore, 2000).
S. Asmussen, S. Foss and D. Korshunov, Asymptotics for sums of random variables with local subexponential behaviour, J. Theoret. Probab. 16 (2003) 489–518.
N.H. Bingham, C.M. Goldie and J.L. Teugels, Regular Variation (Cambridge Univ. Press, Cambridge, 1987).
A.A. Borovkov, Large deviations probabilities for random walks in the absence of finite expectations of jumps, Probab. Theory Related Fields 125 (2003) 421–446.
V.P. Chistyakov, A theorem on sums of independent random positive variables and its applications to branching processes, Theory Probab. Appl. 9 (1964) 710–718.
E.B. Dynkin, Some limit theorems for sums of independent random variables with infinite mathematical expectations, Select. Transl. Math. Statist. Probab. 1 (1961) 171–189.
P. Embrechts, C. Klüppelberg and T. Mikosch, Modelling Extremal Events (Springer, Berlin, 1997).
P. Embrechts and E. Omey, A property of longtailed distributions, J. Appl. Probab. 21 (1984) 80–87.
P. Embrechts and N. Veraverbeke, Estimates for the probability of ruin with special emphasis on the possibility of large claims, Insurance Math. Economics 1 (1982) 55–72.
K.B. Erickson, Strong renewal theorems with infinite mean, Trans. Amer. Math. Soc. 151 (1970) 263–291.
K.B. Erickson, The strong law of large numbers when the mean is undefined, Trans. Amer. Math. Soc. 185 (1973) 371–381.
W. Feller, An Introduction to Probability Theory and Its Applications, Vol. 2, 2nd ed. (Wiley, New York, 1971).
C. Klüppelberg, Subexponential distributions and integrated tails, J. Appl. Probab. 25 (1988) 132–141.
C. Klüppelberg, A.E. Kyprianou and R.A. Maller, Ruin probabilities and overshoots for general Lévy insurance risk processes (2003) submitted.
D. Korshunov, On distribution tail of the maximum of a random walk, Stochastic Process. Appl. 72 (1997) 97–103.
D.A. Korshunov, Large-deviation probabilities for maxima of sums of independent random variables with negative mean and subexponential distribution, Theory Probab. Appl. 46 (2002) 355–366.
J.L. Teugels, The class of subexponential distributions, Ann. Probab. 3 (1975) 1000–1011.
N. Veraverbeke, Asymptotic behavior of Wiener-Hopf factors of a random walk, Stochastic Process. Appl. 5 (1977) 27–37.
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Denisov, D., Foss, S. & Korshunov, D. Tail Asymptotics for the Supremum of a Random Walk when the Mean Is not Finite. Queueing Systems 46, 15–33 (2004). https://doi.org/10.1023/B:QUES.0000021140.87161.9c
Issue Date:
DOI: https://doi.org/10.1023/B:QUES.0000021140.87161.9c