Given a markov chain $(X_n)_{n\ge0}$ with transition matrix $P$ I just proved that $(Z_n)_{n\ge0}:=(X_{kn})_{n\ge0}$ is a Markov chain with transition matrix $P^k$, $k\ge1$.
Does this also work for $(Z_n)_{n\ge0}:=(X_{2n+1})_{n\ge0}$?
In case $(Z_n)_{n\ge0}:=(X_{2n+1})_{n\ge0}$ isn't a Markov chain: why not?
And in case it is, how can I express its transition matrix using $P$?