If $\{X_n\}_{n\in\mathbb N_0}$ is a Markov chain is $T:=\{\inf n\ge1:X_{n}=X_{n-1}\}$ a stopping time ?
$\{T=n\}=\{X_0\neq X_1, X_1\neq X_2\dots X_{n-2}\neq X_{n-1},X_{n-1}=X_n\}$, I would say yes, and what if I change the indices in the set by one, i.e. $\tilde T:=\{\inf n\ge1:X_{n}=X_{n+1}\}$, then information up to time $n+1$ is necessary, but is not $T-1=\tilde T$ ?