If we have a Markov chain represented by the matrix $W_{ij}$ and we know it satisfies the detailed balance condition $W_{ij} P_j = W_{ji} P_i$ $\forall i,j$ , then we know that for larger time $n$ the chain $W^n$ will also satisfy the detailed balance condition.
But it is true the inverse?
More precisely: if I know $W_{ij}$ does NOT satisfy the detailed balance condition, what can I say about $W^n$? It is possible that for some n it does hold true that $(W^n)_{ij} P_j = (W^n)_{ji} P_i$ $\forall i,j$ even if $W_{ij} P_j \neq W_{ji} P_i$ $\forall i,j$.
To make sure, I only consider aperiodic irreducible Markov chains as $W_{ij}$.
Thank you very much.