Exact Coupling of Random Walks on Polish Groups

James T. Murphy ()
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James T. Murphy: The University of Texas at Austin

Journal of Theoretical Probability, 2019, vol. 32, issue 4, 1729-1745

Abstract: 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 $$\mu $$ μ started at 0 admits a successful exact coupling with a version $$S^x$$ S x started at x if and only if there is $$n\geqslant 1$$ n ⩾ 1 with $$\mu ^{n} \wedge \mu ^{n}(x+\cdot ) \ne 0$$ μ n ∧ μ n ( x + · ) ≠ 0 . Moreover, when a successful exact coupling exists, the total variation distance between $$S_n$$ S n and $$S^x_n$$ S n x is determined to be $$O(n^{-1/2})$$ O ( n - 1 / 2 ) if x has infinite order, or $$O(\rho ^n)$$ O ( ρ n ) for some $$\rho \in (0,1)$$ ρ ∈ ( 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 $${\mathbb {R}}$$ 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; 60G50; 60F99; 28C10 (search for similar items in EconPapers)
Date: 2019
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DOI: 10.1007/s10959-018-0856-7

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