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The hitting and mixing times are two fundamental quantities associated withMarkov chains. In Peres and Sousi[PS2015] and Oliveira[Oli2012], the authorsshow that the mixing times and "worst-case" hitting times of reversible Markovchains on finite state spaces are equal up to some universal multiplicativeconstant. We use tools from nonstandard analysis to extend this result toreversible Markov chains on general state spaces that satisfy the strong Fellerproperty. Finally, we show that this asymptotic equivalence can be used to findbounds on the mixing times of a large class of Markov chains used in MCMC, suchas typical Gibbs samplers and Metroplis-Hastings chains, even though theyusually do not satisfy the strong Feller property.

Mixing Times and Hitting Times for General Markov Processes