A renewal theorem in the finite-mean case

Author:
J. L. Geluk

Journal:
Proc. Amer. Math. Soc. **125** (1997), 3407-3413

MSC (1991):
Primary 60K05

DOI:
https://doi.org/10.1090/S0002-9939-97-03955-5

MathSciNet review:
1403127

Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: Let $F(.)$ be a c.d.f. on $(0,\infty )$ such that $\bar {F}(.) \equiv 1-F(.)$ is regularly varying with exponent $-\alpha ,~1<\alpha <2$. Then $U(t)- \frac {t}{\mu } -\frac {1}{\mu ^2} \int _0^t \int _s^\infty \bar {F} (v) dv ds = O(t^4 \bar {F}(t)^2 \bar {F}(t^2\bar {F}(t)))$ as $t \to \infty$, where $U(t)=EN(t)$ is the renewal function associated with $F(t)$. Moreover similar estimates are given for distributions in the domain of attraction of the normal distribution and for the variance of $N(t).$ The estimates improve earlier results of Teugels and Mohan.

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Additional Information

**J. L. Geluk**

Affiliation:
Econometric Institute, Erasmus University Rotterdam, P.O. Box 1738, NL-3000 DR Rotterdam, The Netherlands

Email:
jgeluk@few.eur.nl

Keywords:
Renewal function,
regular variation,
key renewal theorem,
domain of attraction

Received by editor(s):
March 12, 1996

Received by editor(s) in revised form:
June 21, 1996

Communicated by:
Stanley Sawyer

Article copyright:
© Copyright 1997
American Mathematical Society