In statistical mechanics, ergodicity means that the average time (infinitely long) is equal to the average of the whole. For Markov chains, it is defined as recurrent and aperiodic positive. Are the two definitions equivalent to the Markov chain?
If the chain is irreducible, then there is a unique limiting distribution and the average time must be the same as the ensemble average. However, do we really need the aperiodicity here? If the chain is aperiodic, there is no limiting distribution, but, as I know, the average time in a periodic chain must converge to the average over the stationary distribution. So I think that "recurrent irreducible and positive" would be enough for the equivalence of the two averages. Is this correct?