Exact Coupling of Random Walks on Polish Groups

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.

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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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Correspondence to James T. Murphy III.

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Murphy, J.T. Exact Coupling of Random Walks on Polish Groups. J Theor Probab 32, 1729–1745 (2019). https://doi.org/10.1007/s10959-018-0856-7

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Keywords

  • Random walk
  • Successful exact coupling
  • Polish group

Mathematics Subject Classification (2010)

  • 60G50
  • 60F99
  • 28C10