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Last exit times for random walks

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  • Doney, R. A.

Abstract

If{Sn, n [greater-or-equal, slanted] 0} is a random walk which drifts to +[infinity], a last exit occurs at (n, Sn) if Sm > Sn for allm> n. In analogy with the more familiar first exits, it is shown that the set of all such (n, Sn) forms a modified two dimensional renewal process on [0, [infinity]) x (-[infinity], [infinity]). Furthermore the interpoint distribution is the same as in the first exit case, and the projection onto the time axis is stationary. The asymptotic behaviour of the distributions of the time at which the kth last exit occurs and the time at which the last exit from (-[infinity], a] occur are given (for fixed k and a respectively) whenever the left-hand tail of the step distribution is either regularly varying or obeys a Cramér-type condition.

Suggested Citation

  • Doney, R. A., 1989. "Last exit times for random walks," Stochastic Processes and their Applications, Elsevier, vol. 31(2), pages 321-331, April.
    • Handle: RePEc:eee:spapps:v:31:y:1989:i:2:p:321-331
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    Cited by:

    1. Landriault, David & Li, Bin & Lkabous, Mohamed Amine & Wang, Zijia, 2023. "Bridging the first and last passage times for Lévy models," Stochastic Processes and their Applications, Elsevier, vol. 157(C), pages 308-334.

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