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One-shot coupling for certain stochastic recursive sequences

Author

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  • Roberts, Gareth O.
  • Rosenthal, Jeffrey S.

Abstract

We consider Markov chains {[Gamma]n} with transitions of the form [Gamma]n=f(Xn,Yn)[Gamma]n-1+g(Xn,Yn), where {Xn} and {Yn} are two independent i.i.d. sequences. For two copies {[Gamma]n} and {[Gamma]n'} of such a chain, it is well known that provided E[log(f(Xn,Yn))] is weak convergence. In this paper, we consider chains for which also [Gamma]n-[Gamma]n'-->0, where · is total variation distance. We consider in particular how to obtain sharp quantitative bounds on the total variation distance. Our method involves a new coupling construction, one-shot coupling, which waits until time n before attempting to couple. We apply our results to an auto-regressive Gibbs sampler, and to a Markov chain on the means of Dirichlet processes.

Suggested Citation

  • Roberts, Gareth O. & Rosenthal, Jeffrey S., 2002. "One-shot coupling for certain stochastic recursive sequences," Stochastic Processes and their Applications, Elsevier, vol. 99(2), pages 195-208, June.
    • Handle: RePEc:eee:spapps:v:99:y:2002:i:2:p:195-208
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    References listed on IDEAS

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    1. Roberts, G. O. & Tweedie, R. L., 1999. "Bounds on regeneration times and convergence rates for Markov chains," Stochastic Processes and their Applications, Elsevier, vol. 80(2), pages 211-229, April.
    2. Elton, John H., 1990. "A multiplicative ergodic theorem for lipschitz maps," Stochastic Processes and their Applications, Elsevier, vol. 34(1), pages 39-47, February.
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    Cited by:

    1. A. Beskos & G. O. Roberts, 2005. "One-Shot CFTP; Application to a Class of Truncated Gaussian Densities," Methodology and Computing in Applied Probability, Springer, vol. 7(4), pages 407-437, December.
    2. Jovanovski, Oliver, 2014. "Convergence bound in total variation for an image restoration model," Statistics & Probability Letters, Elsevier, vol. 90(C), pages 11-16.

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