## A renewal theorem in the finite-mean case

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- by J. L. Geluk PDF
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**125**(1997), 3407-3413 Request permission

## 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.## References

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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
- Received by editor(s): March 12, 1996
- Received by editor(s) in revised form: June 21, 1996
- Communicated by: Stanley Sawyer
- © Copyright 1997 American Mathematical Society
- 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