$\psi$ irreducibility and ergodicity of Markov Processes

Question

How is Markov chain splitting technique useful for inferring ergodicity of a Markov Chain?Assume that I am working with general state space (uncountable say $R^{N}$ but time is discrete. I want to show that the Markov Process is ergodic. I guess that it suffices to show that it is Harris recurrent. To show Harris recurrence I guess that it suffices to show there exists an atom (obtained via splitting the chain after using minorization criteria) the return time (or hitting time) to which has finite mean.

Answer 1

Harris recurrence is not enough. Just like in the discrete setting, you need an additional hypothesis of aperiodicity (close to the same notion for a countable state space). And also, you will need positive Harris-recurrence (i.e., that the invariant measure is positive, not trivial). The main reference for this kind of problem is the book from Meyn and Tweedie "Markov Chains and Stochastic Stability". It is quite complete and yet easy to read.