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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!

Nikola
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  • Use \inf(A) for $\inf(A)$. – parsiad Aug 28 '19 at 08:45
  • Also, you should use the align* environment and \coloneqq instead of := when writing $\LaTeX$. However, \coloneqq is not supported here, so consider using \equiv instead. – parsiad Aug 28 '19 at 09:00

1 Answers1

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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 $V_i=\tilde V_i+1$.

Kavi Rama Murthy
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