Comment by gh02t

Yeah, the hard part of MCMC is going from finite state and time spaces to continuous. But the general concepts and intuition for finite state/time get a LOT more complicated, even though they are ultimately fairly intuitive generalizations of the finite case. Maybe this is a hot take, but I did do my PhD on MCMC and to be honest I think most of the details for the more general case are superfluous for a lot of people who just want to use MCMC, since it boils down to "it basically just works algorithmically the same way as the easier to understand case of finite state/time but beware the actual details of proving that get very involved." The conclusion of the theory for MCMC is that it ends up being pretty darn lenient in terms of caveats and edge cases, so in practice as a user there aren't that many gotchas to not being really well versed in the deepest parts of the theory.

A counterpoint is that finding texts and references that do handle the more advanced cases in depth is quite difficult, bordering on impossible if you want something approachable. I recreated the general proofs for at least the continuous state case as a chapter in my dissertation and stringing together an end-to-end proof with the proper references required quite a lot of digging and discussions with my (co-)advisor.