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Rates of convergence for renewal sequences in the null-recurrent case

Part of: Special processes Markov processes

Published online by Cambridge University Press:  09 April 2009

Richard Isaac
Richard Isaac
Affiliation:
Department of Mathematics and Computer Science, Lehman College, CUNY Bronx, New York 10468, U.S.A.
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Abstract

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Motivated by work of Garsia and Lamperti we consider null-recurrent renewal sequences with a regularly varying tail and seek information about their rate of convergence to zero. The main result shows that such sequences subject to a monotonicity condition obey a limit law whatever the value of the exponent α is, 0 < α < 1. This monotonicity property is seen to hold for a large class of renewal sequences, the so-called Kaluza sequences. This class includes moment sequences, and therefore includes the sequences generated by reversible Markov chains. Several subsidiary results are proved.

MSC classification

Secondary: 60K05: Renewal theory 60J10: Markov chains (discrete-time Markov processes on discrete state spaces)
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 45 , Issue 3 , December 1988 , pp. 381 - 388
Copyright
Copyright © Australian Mathematical Society 1988

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