 Solution

The Markov property and conditioning on the intermediate state give, for the full history through time $2n$,Condition this identity again on the smaller history $(Y_0,\dots ,Y_n)$. Its right-hand side is already determined by $Y_n=X_{2n}$, so the tower property givesThus the sampled sequence is a time-homogeneous Markov chain with transition matrix $P^2$. Checking only its one-time marginal distributions would not by itself establish the Markov property.

It is the every-second-step case of a subsampled Markov chain.