Abstract
AbstractYuval Peres and Perla Sousi showed that the mixing times and average mixing times of reversible Markov chains on finite state spaces are equal up to some universal multiplicative constant. We use tools from nonstandard analysis to extend this result to reversible Markov chains on compact state spaces that satisfy the strong Feller property.
Highlights
Consider the simple random walk on Z n, given by m (.)Xm = δi, modulo n i=for an i.i.d. ∼ sequence δ, δ, . . . of random variables with Unif({−, + }) distribution
For T ∼ Geom(Cn ) for a sufficiently large constant C >, one can check that the distribution of XT is very close to uniform on Z n
Unif({−, + })). is chain is close to uniform a er Θ(n ) steps, but it is natural to ask if smaller modifications can eliminate periodic behaviour; for the above example, choosing T ∼ Unif({Cn, Cn + }) spreads our random time over only two choices but still works well. is minimal modification turns out to work quite generally, and [PS ] shows that this gives an equivalent reduction in the time a discrete chain takes to mix. e modest goal of this paper is to give a quick proof of the analogous result for continuous chains
Summary
For an i.i.d. ∼ sequence δ , δ , . . . of random variables with Unif({− , + }) distribution. Is chain is close to uniform a er Θ(n ) steps, but it is natural to ask if smaller modifications can eliminate periodic behaviour; for the above example, choosing T ∼ Unif({Cn , Cn + }) spreads our random time over only two choices but still works well. Is minimal modification turns out to work quite generally, and [PS ] shows that this gives an equivalent reduction in the time a discrete chain takes to mix (see refinements in [HP ]). Beyond providing a proof of this useful result, we were motivated to write this paper as a way to illustrate how the machinery developed in [ADS ], which shows that mixing times and hitting times are equal up to multiplicative constants for general. Markov processes satisfying regularity conditions, can be used to give fairly quick and simple translations of facts about discrete chains into facts about continuous chains
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