I try to understand the main concept in the Ergodic theorem in both stationary and non-stationary cases. In the stationary case, I know that if the chain is irreducible and aperiodic, it is Ergodic. But in the non-stationary case, i can not comprehend the content deeply. I have read in a article that if a non-stationary Markov chain satisfies following condition, it is Ergodic.
$$\lim_{n \rightarrow \infty} \sup_{v(0)}||v(m,n)-q ||=0$$
where $v(0)$ is the state distribution vector at time $0$, $v(m,n)$ is the state distribution vector after $n$ transitions given the state distribution vector and transition matrix at time $m$ and $q$ is a fixed vector and satisfies following condition. $$ q_i\ge 0, \quad\quad \sum_{i}q_i=1$$
I want to know if above equation implies the mentioned concepts(irreducible and aperiodic)?