$\lim_{n\to \infty} p^{(n)}$ in a homogeneous discrete-time Markov chain

Question

In a homogeneous discrete-time Markov chain with transition matrix $p$, for any states $j$ and $i$

• $T_j$ is the first return time to state $j$ (the "hitting time").

• $M_j$ is the mean recurrence time at state $j$: $$ M_j := E[T_j]=\infty \cdot P(T_j=\infty) + \sum_{n=1}^{\infty} n\cdot P(T_j=n) \, $$

• $T_{ij}$ is the hitting time of $j$ from $i$

• $f_{ij}$ is the probability that the chain ever visits state $j$ if it starts at $i$. Q: If I understand correctly, $f_{ij} = P(T_{ij} < \infty)$?

From Wikipedia,

1. If a state $j$ aperiodic, then $$ \lim_{n \to \infty} (p^{(n)})_{ij} = \frac{C}{M_j}. $$

   where $C$ is the normalizing constant,

   • In Sheldon Ross's Stochastic Processes, Theorem 4.3.1 states the same except that $C$ is $1$. So I wonder if $C$ can be replaced by $1$ in Wikipedia?

2. If a state $j$ aperiodic, then for any other state $i$, $$ \lim_{n \to \infty} (p^{(n)})_{ij} = C \frac{f_{ij}}{M_j}. $$

   • I also wonder if $C$ can be replaced with $1$?

   • When will $f_{ij} = 1$? The reason for this question is to see when will $$ \lim_{n \to \infty} (p^{(n)})_{ij} = \lim_{n \to \infty} (p^{(n)})_{jj}$$ which is necessary for existence of the limiting distribution (same for all initial distributions).

     For example, if the Markov chain is irreducible and positive recurrent (and aperiodic?), will $f_{ij} = 1$?

Thanks and regards!