Some relations between harmonic renewal measures and certain first passage times

Gerold Alsmeyer

Statistics & Probability Letters, 1991, vol. 12, issue 1, 19-27

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.

Keywords: Random; walk; first; passage; times; harmonic; renewal; measure; Spitzer's; identities; Wiener-Hopf; factorization (search for similar items in EconPapers)
Date: 1991
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Citations: View citations in EconPapers (4)

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