  The hitting and mixing times are two fundamental quantities associated with
Markov chains. In Peres and Sousi[PS2015] and Oliveira[Oli2012], the authors
show that the mixing times and "worst-case" hitting 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 general state spaces that satisfy the strong Feller
property. Finally, we show that this asymptotic equivalence can be used to find
bounds on the mixing times of a large class of Markov chains used in MCMC, such
as typical Gibbs samplers and Metroplis-Hastings chains, even though they
usually do not satisfy the strong Feller property.
