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Proceedings of the American Mathematical Society

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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
MathSciNet review: 1403127
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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

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