Lemma

Let $i\in S$. Then $i$ is aperiodic if and only if there exists an integer $M>0$ such that $$P_{ii}^{(n)}>0\mbox{ for each } n\ge M$$

Lemma

Let $n_{1},\cdots$ be a sequence of positive integers with gcd of 1 (i.e. aperiodic). There exists a finite subset $b_{1},\cdots,b_{r}$ and integer $M>0$ such that for any integer $n\ge M$, it can be written as $$n=d_{1}b_{1}+\cdots+d_{r}b_{r}$$where $d_{1},\cdots ,d_{r}$ are non-negative integers.

Lemma (Aperiodicity is a class property)

If $P$ is irreducible and has an aperiodic state $i\in S$, then for any $j,k\in S$, there exists an integer $M>0$, $$P_{jk}^{(n)}>0,\mbox{ for each }n\ge M$$In particular, all states are aperiodic.