# Tail Asymptotics for the Supremum of a Random Walk when the Mean Is not Finite

- 162 Downloads
- 25 Citations

## Abstract

We consider the sums *S*_{ n }=ξ_{1}+⋯+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}.

## Preview

Unable to display preview. Download preview PDF.

### References

- [1]S. Asmussen,
*Applied Probability and Queues*(Wiley, Chichester, 1987).Google Scholar - [2]S. Asmussen,
*Ruin Probabilities*(World Scientific, Singapore, 2000).Google Scholar - [3]S. Asmussen, S. Foss and D. Korshunov, Asymptotics for sums of random variables with local subexponential behaviour, J. Theoret. Probab. 16 (2003) 489–518.Google Scholar
- [4]N.H. Bingham, C.M. Goldie and J.L. Teugels,
*Regular Variation*(Cambridge Univ. Press, Cambridge, 1987).Google Scholar - [5]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.Google Scholar
- [6]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.Google Scholar
- [7]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.Google Scholar
- [8]P. Embrechts, C. Klüppelberg and T. Mikosch,
*Modelling Extremal Events*(Springer, Berlin, 1997).Google Scholar - [9]P. Embrechts and E. Omey, A property of longtailed distributions, J. Appl. Probab. 21 (1984) 80–87.Google Scholar
- [10]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.Google Scholar
- [11]K.B. Erickson, Strong renewal theorems with infinite mean, Trans. Amer. Math. Soc. 151 (1970) 263–291.Google Scholar
- [12]K.B. Erickson, The strong law of large numbers when the mean is undefined, Trans. Amer. Math. Soc. 185 (1973) 371–381.Google Scholar
- [13]W. Feller,
*An Introduction to Probability Theory and Its Applications*, Vol. 2, 2nd ed. (Wiley, New York, 1971).Google Scholar - [14]C. Klüppelberg, Subexponential distributions and integrated tails, J. Appl. Probab. 25 (1988) 132–141.Google Scholar
- [15]C. Klüppelberg, A.E. Kyprianou and R.A. Maller, Ruin probabilities and overshoots for general Lévy insurance risk processes (2003) submitted.Google Scholar
- [16]D. Korshunov, On distribution tail of the maximum of a random walk, Stochastic Process. Appl. 72 (1997) 97–103.Google Scholar
- [17]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.Google Scholar
- [18]J.L. Teugels, The class of subexponential distributions, Ann. Probab. 3 (1975) 1000–1011.Google Scholar
- [19]N. Veraverbeke, Asymptotic behavior of Wiener-Hopf factors of a random walk, Stochastic Process. Appl. 5 (1977) 27–37.Google Scholar