Let us have the following definitions for a Markov Chain $(X_n)_{n \ge 0}$
$$ T_i := inf\{n \ge 1 : X_n = i\}\\ T_i^{(r+1)} := inf\{n \ge T_i^{(r)}+1 : X_n = i\}\\ \tilde V_i := inf\{n \ge 1 : T_i^{(n)} = \infty \} \\ V_i := \sum_{n=0}^{\infty}\unicode{x1D7D9}_{\{X_n=i\}} $$
My question is, are $\tilde V_i$ and $V_i$ the same, how do I prove that? The way I see it, $V_i$ counts the number of visits, and $\tilde V_i$ represents the time at which we no longer visit state $i$. So if $\tilde V_i=2$ shouldn't $V_i=1$?
Thanks in advance!
It is usually assumed that $X_0=i$ (and you look for the number of times the chain returns to $i$). In this case the starting term in $V_i$ is $1$ and we have $V_i=\tilde V_i+1$.