A Question Regarding Markov Chains and Ergodicity

Question

Suppose the Markov chain with Probability Transition Matrix, $P$ = ($p{_x}{_y}$) is ergodic and $p{_m}(x, y) > 0$ for all states $x$ and $y$.

If $n ≥ m$, show that $p_n(x, y) > 0$ for all states $x$ and $y$.

How would I go about doing this?

Answer 1

Did you prove that the transition probability for any markov chain is related to its transition matrix?

If, $P^m>0$ obviously $P^n>0$ for all $n\geq m$..? This can be shown as an ergodic markov chain is irreducible. "Did" implied you to use Chapman-Kolmogorov. Let, $ p_{zy}^{(m)}>0 $. Then there exists $ p_{xz}^{(n-m)}>0 $ due to irreducibility.

Therefore, $ p_{xy}^{(n)} = \sum\limits_{k\in E} p_{xk}^{(n-m)} p_{ky}^{(m)} \geq p_{xz}^{(n-m)}p_{zy}^{(m)} > 0 $