Assuming both $X \to Y \to Z$ and $Y \to Z \to W$ are Markov chains, I want to prove that $X \to Y \to Z \to W$. I don't really know how to do that. The normal formula is:
$P(x,y,z,w) = P(x) P(y|x) P(z|x,y) P(w|x,y,z)$
Now, because $X \to Y \to Z$ is a Markov chain, I know that I can replace $P(z|x,y)$ with $P(z|y)$ and because $Y \to Z \to W$ is a Markov chain, I know that I can replace $P(w|x,y,z)$ with $P(z|x,z)$. But how can I get rid of the $x$?
As stated, this is false.
Let $X,Y,Z$ be i.i.d. and let $W=X$.