Let matrix $A$ be a primitive matrix so $A^k>0$ and $A$ is also a transition (stochastic) matrix . Can we say that $A$ is ergodic?In other words, can we say that $A$ is:
(1) strongly connected
(2)aperiodic?
The first condition (1) is trivial by the definition of primitive. But regarding the second (2) condition I am not sure. My intuition comes from the lazy random walk concept that $p(I+A)$ is ergodic where $p\in[0,1]$ and usually set to $1/2$.
Can you help me to formalize that? I couldn't find any formal definition online.
The condition $A^n>0$ for some $n$ does imply aperiodicity. There must be a pair of states $i,j$ with $A_{ij}>0$, in which case $A^{n+1}_{ij} \geqslant (A^n_{ii})(A_{ij})>0$, so that $$ \gcd \{k:A^k_{ij}>0\}=\gcd\{n,n+1\}=1. $$ Hence $i$ and $j$ are aperiodic, and periodicity is a class property, so as $A$ is irreducible, it follows that $A$ is aperiodic.