I have the following sequence of random variables,$\{\mathbf{X}_{n}\}_{n\in\mathbb{N}}$, defined recursively by
$$ \mathbf{X}_{n+1} = \mathbf{\Phi}\mathbf{X}_{n} + \mathbf{\Psi}_{z}\mathbf{Z}_{n} + \mathbf{\Psi}_{w}\mathbf{W}, $$
where $\{\mathbf{Z}_{n}\}_{n\in\mathbb{N}}$ is a (bivariate) Markov chain and $\{\mathbf{W}_{n}\}_{n\in\mathbb{N}}$ is (bivariate) i.i.d. (and normally distributed) noise, thus also a Markov chain. Furthermore, $\mathbf{\Phi}$, $\mathbf{\Psi}_{z}$ and $\mathbf{\Psi}_{w}$ are constant (matrices). Since each $\mathbf{X}_{n}$ for $n\in\mathbb{N}^{\ast}$ only depends on the previous state, i.e. $\mathbf{X}_{n-1}$, this makes it a Markov chain? We do not have to regard $\mathbf{Z}_{n}$ and $\mathbf{W}_{n}$ since these are already Markov, am I correct in my assessment?
I found on this page about AutoRegressive processes that the AR(1) process is a Markov chain given that the noise is mutually independent, which seems to apply to my case, although I have the process $\{\mathbf{Z}_{n}\}_{n\in\mathbb{N}}$ involved also.
Edit: possible solution I came up with (inspired by this post) is that $X_{n}$ only depends, as seen by the recursion, of $X_{n-1}$. Thus, the probability distribution of $X_{n}$ conditional on its history reduces to simply the conditional of the prior state $X_{n-1}$.