Hitting, return, and visit times

Let $(X_n,{\mathcal{F}}_n{)}_{n\ge 0}$ be an adapted Stochastic processes and their finite-dimensional distributions with values in a countable set $E$ equipped with $2^E$. For $A\subseteq E$, define

$T_A:=\inf \{n\ge 0:X_n\in A\},T_x:=T_{\{x\}},T_x^{+}:=\inf \{n\ge 1:X_n=x\}\mathrm{.}$

The infimum of the empty set is $+\mathrm{\infty }$. The visit count is the extended nonnegative integer

$N_x:={\sum }_{n\ge 0}{\mathbf{1}}_{\{X_n=x\}}\in {\mathbb{N}}_0\cup \{+\mathrm{\infty }\}\mathrm{.}$

For a process started at $x$, put $R_0=0$ and define recursively

$R_k=\{\begin{array}{ll}\inf \{n>R_{k-1}:X_n=x\},& R_{k-1}<\mathrm{\infty },\\ +\mathrm{\infty },& R_{k-1}=\mathrm{\infty }\mathrm{.}\end{array}$

Thus a later return time is assigned $+\mathrm{\infty }$ if the preceding one is infinite; the expression $X_{\mathrm{\infty }}$ is never used. For every $A$ and $n\ge 0$,

$\{T_A\le n\}={\bigcup }_{j=0}^n\{X_j\in A\},\{T_x^{+}\le n\}={\bigcup }_{j=1}^n\{X_j=x\},$

where the second union is empty when $n=0$. Adaptedness makes these events belong to ${\mathcal{F}}_n$, so these are stopping times under Discrete stopping time.