Let $P$ be a probability distribution of an ergodic Markov chain on the space $\Omega_{n}=\left\{\omega=\omega_{1},\ldots,\omega_{n}\right\}$ with finite state space $X$. Then, for any $x \in X$, we have that $P(\omega: \omega_{i} \neq x, 1 \leq i \leq n) \leq C q^{n},$ for some constant $C$ and $0 < q <1$.
Is there any nice way of doing this? Thanks!