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Bounds on Lifting Continuous-State Markov Chains to Speed Up Mixing

Author

Listed:
  • Kavita Ramanan

    (Brown University)

  • Aaron Smith

    (University of Ottawa)

Abstract

It is often possible to speed up the mixing of a Markov chain $$\{ X_{t} \}_{t \in \mathbb {N}}$$ { X t } t ∈ N on a state space $$\Omega $$ Ω by lifting, that is, running a more efficient Markov chain $$\{ \widehat{X}_{t} \}_{t \in \mathbb {N}}$$ { X ^ t } t ∈ N on a larger state space $$\hat{\Omega } \supset \Omega $$ Ω ^ ⊃ Ω that projects to $$\{ X_{t} \}_{t \in \mathbb {N}}$$ { X t } t ∈ N in a certain sense. Chen et al. (Proceedings of the 31st annual ACM symposium on theory of computing. ACM, 1999) prove that for Markov chains on finite state spaces, the mixing time of any lift of a Markov chain is at least the square root of the mixing time of the original chain, up to a factor that depends on the stationary measure of $$\{X_t\}_{t \in \mathbb {N}}$$ { X t } t ∈ N . Unfortunately, this extra factor makes the bound in Chen et al. (1999) very loose for Markov chains on large state spaces and useless for Markov chains on continuous state spaces. In this paper, we develop an extension of the evolving set method that allows us to refine this extra factor and find bounds for Markov chains on continuous state spaces that are analogous to the bounds in Chen et al. (1999). These bounds also allow us to improve on the bounds in Chen et al. (1999) for some chains on finite state spaces.

Suggested Citation

  • Kavita Ramanan & Aaron Smith, 2018. "Bounds on Lifting Continuous-State Markov Chains to Speed Up Mixing," Journal of Theoretical Probability, Springer, vol. 31(3), pages 1647-1678, September.
  • Handle: RePEc:spr:jotpro:v:31:y:2018:i:3:d:10.1007_s10959-017-0745-5
    DOI: 10.1007/s10959-017-0745-5
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    References listed on IDEAS

    as
    1. Yuval Peres & Perla Sousi, 2015. "Mixing Times are Hitting Times of Large Sets," Journal of Theoretical Probability, Springer, vol. 28(2), pages 488-519, June.
    2. Alison L. Gibbs & Francis Edward Su, 2002. "On Choosing and Bounding Probability Metrics," International Statistical Review, International Statistical Institute, vol. 70(3), pages 419-435, December.
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