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Submultiplicative moments of the supremum of a random walk with negative drift

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  • Sgibnev, M. S.

Abstract

Let {Sn} be the sequence of partial sums of independent identically distributed random variables with negative mean. Necessary and sufficient conditions are obtained for E[phi](M[infinity]) to be finite, where [phi](x) is a non-decreasing submultiplicative function, i.e. [phi](x + y)[less-than-or-equals, slant][phi](x)[phi](y), and M[infinity] = sup{0, S1, S2,...}. This generalizes a well-known result on moments of M[infinity] proved by Kiefer and Wolfowitz (1956). Submultiplicative moments of the first positive sum are also considered.

Suggested Citation

  • Sgibnev, M. S., 1997. "Submultiplicative moments of the supremum of a random walk with negative drift," Statistics & Probability Letters, Elsevier, vol. 32(4), pages 377-383, April.
    • Handle: RePEc:eee:stapro:v:32:y:1997:i:4:p:377-383
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    References listed on IDEAS

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    1. Embrechts, Paul & Goldie, Charles M., 1982. "On convolution tails," Stochastic Processes and their Applications, Elsevier, vol. 13(3), pages 263-278, September.
    2. 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.
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