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Journal of Theoretical Probability

, Volume 32, Issue 4, pp 1729–1745 | Cite as

Exact Coupling of Random Walks on Polish Groups

  • James T. MurphyIIIEmail author
Article

Abstract

Exact coupling of random walks is studied. Conditions for admitting a successful exact coupling are given that are necessary and in the Abelian case also sufficient. In the Abelian case, it is shown that a random walk S with step-length distribution \(\mu \) started at 0 admits a successful exact coupling with a version \(S^x\) started at x if and only if there is \(n\geqslant 1\) with \(\mu ^{n} \wedge \mu ^{n}(x+\cdot ) \ne 0\). Moreover, when a successful exact coupling exists, the total variation distance between \(S_n\) and \(S^x_n\) is determined to be \(O(n^{-1/2})\) if x has infinite order, or \(O(\rho ^n)\) for some \(\rho \in (0,1)\) if x has finite order. In particular, this paper solves a problem posed by H. Thorisson on successful exact coupling of random walks on \({\mathbb {R}}\). It is also noted that the set of such x for which a successful exact coupling can be constructed is a Borel measurable group. Lastly, the weaker notion of possible exact coupling and its relationship to successful exact coupling are studied.

Keywords

Random walk Successful exact coupling Polish group 

Mathematics Subject Classification (2010)

60G50 60F99 28C10 

Notes

Acknowledgements

This work was supported by a grant of the Simons Foundation (#197982 to The University of Texas at Austin). The author also thanks the anonymous referees for her or his helpful suggestions.

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Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2018

Authors and Affiliations

  1. 1.The University of Texas at AustinAustinUSA

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