Summary
Consider a reversible Markov chain X n which takes values in a subset of ℤd. If the steps of the chain are uniformly bounded and the invariant measure satisfies a mild regularity condition, Varopoulos, Carne and Kesten have obtained estimates on \(P(|X_n - X_{\text{0}} | > \lambda n^{1/2} )\) which exhibit a Gaussian tail in λ but blow up as n→∞. Following Kesten's approach we derive bounds which are uniform in n in some special cases. Our main result, however, is an example which shows that in general the estimates of Varopoulos, Carne and Kesten are essentially best possible.
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Research partially supported by an S.E.R.C. (U.K.) visiting fellowship and an operating grant from N.S.E.R.C. of Canada
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Barlow, M.T., Perkins, E.A. Symmetric Markov chains in ℤd: How fast can they move?. Probab. Th. Rel. Fields 82, 95–108 (1989). https://doi.org/10.1007/BF00340013
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DOI: https://doi.org/10.1007/BF00340013