The hypotheses you have now don't seem to rule out that here are 2 states $i\ne j$ with $P_{i,j}=1$ and $P_{j,i}=1$. In this case $P^n$ does not converge because $P^n_{i,i}$ is 0 or 1 depending on whether $n$ is odd or even.
Bjørn Kjos-Hanssen
- 24.8k
- 3
- 58
- 114