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On coupling of random walks and renewal processes

Part of: Special processes

Published online by Cambridge University Press:  14 July 2016

Torgny Lindvall  and
L. C. G. Rogers
Torgny Lindvall*
Affiliation:
University of Göteborg
L. C. G. Rogers*
Affiliation:
University of Bath
*
∗Postal address: Department of Mathematics, University of Göteborg, 41296 Göteborg, Sweden. E-mail: lindvall@math.chalmers.se
∗∗Postal address: School of Mathematical Sciences, University of Bath, Claverton Down, Bath BA2 2AY, U.K. E-mail: l.c.g.rogers@maths.bath.ac.uk
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Abstract

The use of Mineka coupling is extended to a case with a continuous state space: an efficient coupling of random walks S and S' in can be made such that S' — S is virtually a one-dimensional simple random walk. This insight settles a zero-two law of ergodicity. One more proof of Blackwell's renewal theorem is also presented.

Keywords

ERGODICITYRANDOM WALKCOUPLINGBLACKWELL'S RENEWAL THEOREM

MSC classification

Secondary: 60K05: Renewal theory
Type
Research Papers
Information
Journal of Applied Probability , Volume 33 , Issue 1 , March 1996 , pp. 122 - 126
Copyright
Copyright © Applied Probability Trust 1996 

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References

[1] Kerstan, J. and Matthes, K. (1968) Gleichverteilungseigenschaften von Faltungen von Verteilungsgesetzen auf lokalkompakten abelschen Gruppen. I. Math. Nachr. 37, 267–312.Google Scholar
[2] Krengel, U. (1985) Ergodic Theorems. de Gruyter, Berlin.Google Scholar
[3] Lindvall, T. (1992) Lectures on the Coupling Method. Wiley, New York.Google Scholar
[4] Revuz, D. (1984) Markov Chains. 2nd edn. North-Holland, Amsterdam.Google Scholar