This article is cited in 3 papers

Short Communications

Asymptotics of the generalized renewal functions when the variance is finite

M. S. Sgibnev

Sobolev Institute of Mathematics, Siberian Branch of the Russian Academy of Sciences

Abstract: We study the asymptotic behavior, as $t \to\infty$, of the generalized renewal functions
$$ \Phi_n(t)=\sum_{k=0}^\infty\frac{n\cdot(n+k-1)!}{k!}\mathsf{P}\{S_k\le t\}, $$
where $n>0$ is an integer and $S_{k}$ are partial sums of a sequence of independent identically distributed random variables with positive mean and finite variance.

Keywords: generalized renewal functions, higher renewal moments, random walk, ladder epochs.

Received: 05.06.1995
Revised: 02.04.1996

DOI: 10.4213/tvp2007

 English version: Theory of Probability and its Applications, 1998, 42:3, 536–541

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