If a $MC$ is irreducible and aperiodic, for every node pair $(i, j)$, there exists $N(i, j)$ such that $P^{n_{ij}} > 0$ whenever $n\geq N(i, j).$

Question

How can I go about proving the following?

If a $MC$ is irreducible and aperiodic, for every node pair $(i, j)$, there exists $N(i, j)$ such that $P^{n_{ij}} > 0$ whenever $n \geq N(i, j).$

Answer 1

See Proposition 1.7 on page 8 of Markov Chains and Mixing Times by Levin, Peres, and Wilmer.

If $P$ is aperiodic and irreducible, then there is an integer $r$ such that $P^r(x,y) > 0$ for all $x,y ∈ \Omega$.

The text is available for free online (on this site for instance). The result is presented and proved in a straightforward fashion. I suggest that you read through the proof and make a new question post about any questions you have, if you have any.