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Some relations between harmonic renewal measures and certain first passage times

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  • Alsmeyer, Gerold

Abstract

Let X1, X2,... be i.i.d. random variables with common mean [mu] [greater-or-equal, slanted] 0 and associated random walk S0 = 0, Sn = X1 + ... + Xn, n [greater-or-equal, slanted] 1. Let U(t) = [Sigma]n [greater-or-equal, slanted] 1(1/n)P(Sn [less-than-or-equals, slant] t) be the harmonic renewal function of (Sn)n [greater-or-equal, slanted] 0 and [tau](t) = inf{itn [greater-or-equal, slanted] 1: Sn > t}. It is shown that U(t) = E[Psi]([tau](t)) + [gamma] for all t [greater-or-equal, slanted] 0, where [Psi](t) denotes Euler's psi function and [gamma] Euler's constant. This identity is further used to derive a number of interesting global and asymptotic properties of U(t). Some extensions to so-called generalized renewal measures are discussed in the final section.

Suggested Citation

  • Alsmeyer, Gerold, 1991. "Some relations between harmonic renewal measures and certain first passage times," Statistics & Probability Letters, Elsevier, vol. 12(1), pages 19-27, July.
    • Handle: RePEc:eee:stapro:v:12:y:1991:i:1:p:19-27
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    Cited by:

    1. Khaniyev, T. & Kesemen, T. & Aliyev, R. & Kokangul, A., 2008. "Asymptotic expansions for the moments of a semi-Markovian random walk with exponential distributed interference of chance," Statistics & Probability Letters, Elsevier, vol. 78(6), pages 785-793, April.
    2. Khaniyev, Tahir & Kucuk, Zafer, 2004. "Asymptotic expansions for the moments of the Gaussian random walk with two barriers," Statistics & Probability Letters, Elsevier, vol. 69(1), pages 91-103, August.
    3. M. S. Sgibnev, 1998. "On the Asymptotic Behavior of the Harmonic Renewal Measure," Journal of Theoretical Probability, Springer, vol. 11(2), pages 371-382, April.
    4. Mallor, F. & Omey, E. & Santos, J., 2007. "Multivariate weighted renewal functions," Journal of Multivariate Analysis, Elsevier, vol. 98(1), pages 30-39, January.

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