Say I have an irreducible Markov chain with state space $\{1, 2, 3 ... m\}$, where $m > 2$ and stationary distribution $s = \{s_1, s_2, ... s_m\}$. The initial state is given by the stationary distribution, so $P(X_0 = i) = s_i$.
Why is it that all of $X_0, X_1 ... X_n$ have the stationary distribution? Apparently, it's because of how $X_0$ does... how does that work?