Given two Markov chains $X_t$, $Y_t$ characterized by the same transition matrix $P$, let $\tau_c$ be the first time the two chains have the same state, i.e. $\tau_c = \min\{t:X_t=Y_t\}$.
The question is to show $\Bbb P(\tau_c\le t_0) = \Bbb P(\tau_c\le 2t_0|\tau_c > t_0)$. This seems intuitively correct, but I am not sure how to show it. Anyone can help with a proof? Thank you!