An irreducible chain with finitely many states is aperiodic if and only if there exists an n such that $p_{i,j}^{(n)}>0$, $\forall i,j \in S$

Question

An irreducible chain with finitely many states is aperiodic if and only if there exists an n such that $p_{i,j}^{(n)}>0$, $\forall i,j \in S$

I'm trying to prove this, I'm thinking about using this result:

If I is an aperiodic state then there exists $n_0$ such that $p_{ii}^{(n)}>0$, $\forall n\geq n_0$

But I'm a little confused, since the chain is aperiodic $d_i = 1$, and also, there exists $n_0$ such that $p_{ii}^{(n)}>0$, $\forall n\geq n_0$ by the last result.

But how can I infer that this $p_{i,j}^{(n)}>0$ ?

It's a little confused this topic, could someone help me please.

Thanks for your time and help.

Answer 1

Remember a chain is irreducible iff, for every $i,j$, there exists $n=n(i,j)\geq 1$ such that $p_{i,j}^{(n)}>0$.

Now you need to make $n(i,j)$ independent of $i,j$ by aperiodicity. But the chain is finite and we have $p_{i,j}^{(n+m)}\geq p_{i,i}^{(m)} p_{i,j}^{(n)}$, so for $n\geq n_0+\max_{i,j}n(i,j)$ we have ...