Show a markov chain with transition matrix $P$ and a markov chain with matrix $\frac{1}{2}(I+P))$ have the same invariant distribution

Question

Show that if $(X_n)_{n\geq 0}$ is a finite state irreducible Markov chain with transition matrix $P$, then a Markov chain with transition matrix $Q= \frac{1}{2}(I+P)$ is irreducible and aperiodic. Moreover, show that the two chains have the same invariant distribution.

I managed to show that the markov chain with transition matrix $Q$ is irreducible and aperiodic, but I'm stuck at showing they have the same invariant distribution.

I've played around a bit with that $\pi_1P=\pi_1$ and $\pi_2( \frac{1}{2}(I+P))=\pi_2$, where $\pi_1$ and $\pi_2$ are the invariant distributions. But I'm not really getting anywhere.

Answer 1

For $\pi_1$, $$ \begin{aligned} \pi_1P&=\pi_1 \end{aligned} $$

For $\pi_2$, $$ \begin{aligned} \pi_2[\frac{1}{2}(I+P)]&=\pi_2 \\ \pi_2[\frac{1}{2}(I+P)]-\pi_2&=0 \\ \pi_2[\frac{1}{2}(I+P)-I]&=0 \\ \pi_2[\frac{1}{2}(P-I)]&=0\\ \frac{1}{2}\pi_2(P-I)&=0 \\ \pi_2(P-I)&=0\\ \pi_2P-\pi_2&=0\\ \pi_2P&=\pi_2 \end{aligned} $$

I think this can answer your question.