Aperiodic but not irreductible Markov Chain

Question

If I understood well, a Markov Chain with state space $E$ is said to be irreductible if for all $x,y\in E$ there is $n$ such that $$P^n(x,y)>0,$$ where $P$ is the transition matrix.

Also, I know that a Markov chain is aperiodic if and only if for all $x,y\in E$ there is $N$ such that for all $n>N$: $$P^n(x,y)>0.$$

Then it seems clear to me that every aperiodic Markov chain is irreductible. That is, in fact, true? If not, what have I got wrong and what is a counter-example?

Thank you

Answer 1

Your definition of aperiodicity is incorrect and not equivalent to the usual one that $\ \gcd\big\{n\,\big|\,P^n(x,x)>0\,\big\}=1\ $ for all states $\ x\ $. Apart from the problem pointed out by Ilmari Koronen in the comments, you're requiring $\ P^n(x,y)>0\ $ for all $\ x\ $ and $\ y\ $. In the definition of aperiodicity, you take the gcd over only those $\ n\ $ for which $\ P^n(x,\color{red}{x})>0\ $, not over those for which $\ P^n(x,y)>0\ $ where $\ y\ne x\ $.

If $\ P_1, P_2\ $ are transition matrices for two aperiodic Markov chains, then the Markov chain with transition matrix $$ \pmatrix{P_1&0\\0&P_2} $$ is not irreducible, but it will be aperiodic.

Answer 2

Your statement is correct. Aperiodicity implies irreducibility but is stronger. Whereas in the case of irreducibility your integer variable $n$ generally depends on the pair $x,y$ chosen, aperiodicity allows one to choose $n$ independent of any pair $x,y$.