This article is cited in 2 papers

Short Communications

Second order renewal theorem in the finite-means case

A. Baltrūnas^a, E. Omey<sup>b

a</sup> Institute of Mathematics and Informatics
^b Hogeschool-Universiteit Brussel

Abstract: Let $F$ be a distribution function (d.f.) on $(0, \infty )$ and let $U$ be the renewal function associated with $F$. If $F$ has a finite first moment $\mu$, then it is well known that $U(t)$ asymptotically equals $t/\mu$. It is also well known that $U(t)-t/\mu $ asymptotically behaves as $S(t)/\mu, $ where $S$ denotes the integral of the integrated tail distribution $F_1$ of $F$. In this paper we discuss the rate of convergence of $U(t)-t/\mu -S(t)/\mu $ for a large class of distribution functions. The estimate improves earlier results of Geluk, Teugels, and Embrechts and Omey.

Keywords: renewal function, subexponential distributions, regular variation, $O$-regular variation.

Received: 17.12.1999

Language: English

DOI: 10.4213/tvp3381

 English version: Theory of Probability and its Applications, 2003, 47:1, 127–132

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