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Geometric renewal convergence rates from hazard rates

Part of: Special processes Limit theorems

Published online by Cambridge University Press:  14 July 2016

Kenneth S. Berenhaut  and
Robert Lund
Kenneth S. Berenhaut*
Affiliation:
University of Georgia
Robert Lund*
Affiliation:
University of Georgia
*
∗ Postal address: Department of Statistics, University of Georgia, Athens, GA 30602-1952, USA.
∗ Postal address: Department of Statistics, University of Georgia, Athens, GA 30602-1952, USA.
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Abstract

This paper studies the geometric convergence rate of a discrete renewal sequence to its limit. A general convergence rate is first derived from the hazard rates of the renewal lifetimes. This result is used to extract a good convergence rate when the lifetimes are ordered in the sense of new better than used or increasing hazard rate. A bound for the best possible geometric convergence rate is derived for lifetimes having a finite support. Examples demonstrating the utility and sharpness of the results are presented. Several of the examples study convergence rates for Markov chains.

Keywords

geometric convergencerenewal sequenceMarkov chainspower serieshazard ratesincreasing hazard ratenew better than used

MSC classification

Primary: 60K05: Renewal theory
Secondary: 60F99: None of the above, but in this section
Type
Research Papers
Information
Journal of Applied Probability , Volume 38 , Issue 1 , March 2001 , pp. 180 - 194
Copyright
Copyright © by the Applied Probability Trust 2001 

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