In the book "Knowing the Odds", Basic Limit Theorem for Markov Chain is stated as follows.
Theorem 7.41 (Basic Limit Theorem). Suppose $j$ is a recurrent aperiodic state in an irreducible Markov chain. Then for each i, $ \lim_{n\to\infty} P_{ij}^{(n)}= \dfrac{1}{E^{j}\left\{T_{j}\right\}} $.
In the second line of the proof of this theorem, the author mentions that "To simplify the proof, suppose that $P_{jj}>0$, so that $f_{1}>0$. " But what if $P_{jj}=0$? Is it related to the condition "aperiodic" in this theorem? I had checked the rest of the proof, but I don't think aperiodic in the case $P_{jj}>0$ is needed.
Thanks!