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

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.

• $\pi_2( \frac{1}{2}(I+P))=\pi_2 \implies \pi_2 P = \pi_2$, since $P$ is irreductible and aperiodic, the stationnary distribution is unique. Commented May 28, 2019 at 13:59

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}