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Tail expansions for the distribution of the maximum of a random walk with negative drift and regularly varying increments

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  • Barbe, Ph.
  • McCormick, W.P.
  • Zhang, C.

Abstract

Let F be a distribution function with negative mean and regularly varying right tail. Under a mild smoothness condition we derive higher order asymptotic expansions for the tail distribution of the maxima of the random walk generated by F. The expansion is based on an expansion for the right Wiener-Hopf factor which we derive first. An application to ruin probabilities is developed.

Suggested Citation

  • Barbe, Ph. & McCormick, W.P. & Zhang, C., 2007. "Tail expansions for the distribution of the maximum of a random walk with negative drift and regularly varying increments," Stochastic Processes and their Applications, Elsevier, vol. 117(12), pages 1835-1847, December.
    • Handle: RePEc:eee:spapps:v:117:y:2007:i:12:p:1835-1847
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    References listed on IDEAS

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    1. Veraverbeke, N., 1977. "Asymptotic behaviour of Wiener-Hopf factors of a random walk," Stochastic Processes and their Applications, Elsevier, vol. 5(1), pages 27-37, February.
    2. Geluk, J. L., 1992. "Second order tail behaviour of a subordinated probability distribution," Stochastic Processes and their Applications, Elsevier, vol. 40(2), pages 325-337, March.
    3. Asmussen, Søren & Kalashnikov, Vladimir & Konstantinides, Dimitrios & Klüppelberg, Claudia & Tsitsiashvili, Gurami, 2002. "A local limit theorem for random walk maxima with heavy tails," Statistics & Probability Letters, Elsevier, vol. 56(4), pages 399-404, February.
    4. Geluk, J. L., 1996. "Tails of subordinated laws: The regularly varying case," Stochastic Processes and their Applications, Elsevier, vol. 61(1), pages 147-161, January.
    5. Omey, E. & Willekens, E., 1986. "Second order behaviour of the tail of a subordinated probability distribution," Stochastic Processes and their Applications, Elsevier, vol. 21(2), pages 339-353, February.
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    Cited by:

    1. Lin, Jianxi, 2012. "Second order asymptotics for ruin probabilities in a renewal risk model with heavy-tailed claims," Insurance: Mathematics and Economics, Elsevier, vol. 51(2), pages 422-429.

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