Given a state space of $\{1, 2, 3\}$, and probabilities to transition $p(1,2)$, etc (which are given), how would I know if
$Y_n = X_{5n} or Z_n = X_{n+17}$ or $W_n = (X_n)^2$
are Markov chains?
I get the feeling that, for $Y_n = X_{5n}$, I need to take a matrix, and take its $5$th power, but then what? What properties does this 5th-power matrix need to satisfy?