Solution
= Solution
A randomized stopping time $\tau$ is a <strong stationary time> when
$$
\mathbb P_x(X_\tau=y,\tau=t)=\pi(y)\mathbb P_x(\tau=t)
$$
for every initial state $x$, state $y$, and time $t$. Equivalently, $X_\tau$ has distribution $\pi$ and is independent of $\tau$.