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A Note on Veraverbeke's Theorem

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Abstract

We give an elementary probabilistic proof of Veraverbeke's theorem for the asymptotic distribution of the maximum of a random walk with negative drift and heavy-tailed increments. The proof gives insight into the principle that the maximum is in general attained through a single large jump.

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References

  1. S. Asmussen, Ruin Probabilities (World Scientific, Singapore, 2000).

    Google Scholar 

  2. S. Asmussen, H. Schmidli and V. Schmidt, Tail probabilities for non-standard risk and queueing processes with subexponential jumps, Adv. in Appl. Probab. 31 (1999) 422–447.

    Google Scholar 

  3. F. Baccelli and S.G. Foss, Moments and tails in monotone-separable stochastic networks, Ann. Appl. Probab. (2003) to appear.

  4. P. Embrechts, C. Klüppelberg and T. Mikosch, Modelling Extremal Events (Springer, Berlin, 1997).

    Google Scholar 

  5. P. Embrechts and N. Veraverbeke, Estimates for the probability of ruin with special emphasis on the possibility of large claims, Insurance Math. Econom. 1 (1982) 55–72.

    Google Scholar 

  6. S.G. Foss and S. Zachary, Asymptotics for the maximum of a modulated random walk with heavy-tailed increments, in: Analytic Methods in Applied Probability: In Memory of Fridrikh Karpelevich, ed. Yu.M. Suhov (Amer. Math. Soc., Providence, RI, 2002).

    Google Scholar 

  7. V. Kalashnikov and G. Tsitsiashvili, Tails of waiting times and their bounds, Queueing Systems 32 (1999) 257–283.

    Google Scholar 

  8. D.A. Korshunov, On distribution tail of the maximum of a random walk, Stochastic Process. Appl. 72(1) (1997) 97–103.

    Google Scholar 

  9. D.A. Korshunov, Large-deviation probabilities for maxima of sums of independent random variables with negative mean and subexponential distribution, Theory Probab. Appl. 46(2) (2002) 355–366.

    Google Scholar 

  10. K. Sigman, A primer on heavy-tailed distributions, Queueing Systems 33 (1999) 261–275.

    Google Scholar 

  11. N. Veraverbeke, Asymptotic behavior of Wiener-Hopf factors of a random walk, Stochastic Process. Appl. 5 (1977) 27–37.

    Google Scholar 

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Zachary, S. A Note on Veraverbeke's Theorem. Queueing Systems 46, 9–14 (2004). https://doi.org/10.1023/B:QUES.0000021155.44510.9f

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  • DOI: https://doi.org/10.1023/B:QUES.0000021155.44510.9f

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