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A note on the maximum of a random walk

Author

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  • Stadje, W.

Abstract

For an integer-valued, aperiodic random walk (Sn)n [greater-or-equal, slanted] 1 with negative drift we consider its maximum M and M+, its maximum before first attaining a nonpositive value. As a converse to well-known results on the asymptotic behavior of mn = P(M [greater-or-equal, slanted] n) and rn = P(M+ [greater-or-equal, slanted] n), it is shown that any of the relations mn ~ cr-n or rn ~ cr-n (c > 1, r > 1) implies that E(rS1) = 1.

Suggested Citation

  • Stadje, W., 1995. "A note on the maximum of a random walk," Statistics & Probability Letters, Elsevier, vol. 23(3), pages 227-231, May.
    • Handle: RePEc:eee:stapro:v:23:y:1995:i:3:p:227-231
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    Cited by:

    1. Korshunov, D., 1997. "On distribution tail of the maximum of a random walk," Stochastic Processes and their Applications, Elsevier, vol. 72(1), pages 97-103, December.

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