Let $(X_m)$ be a finite space discrete time irreducible and aperiodic Markov chain with stationary distribution $\pi$. The state space is a finite set of positive integers $\{x_1, x_2, \dots, x_l\}$.
What does $\lim\limits_{n \to \infty} \left( \prod\limits_{i=1}^n X_i \right)^{1/n}$ equal to?
Is it true that we get the same limit if we consider just the last $k$ realizations, that is, is it true that $\lim\limits_{k \to \infty} \lim\limits_{n \to \infty} \left( \prod\limits_{i=n-k+1}^n X_i \right)^{1/k}$ defines the same limit? (Does it exist for all irreducible, aperiodic discrete Markov chain or some more assumptions are needed?)