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Proceedings of the American Mathematical Society

Published by the American Mathematical Society, the Proceedings of the American Mathematical Society (PROC) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6826 (online) ISSN 0002-9939 (print)

The 2020 MCQ for Proceedings of the American Mathematical Society is 0.85.

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A renewal theorem in the finite-mean case
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by J. L. Geluk PDF
Proc. Amer. Math. Soc. 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.
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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