State space and transition probability of Markov Chain

Question

Let $X={X_n}$ be a Markov chain with state space $S$ and transition probability matrix $P= [P_{ij}]$.Let $Y$ be a Markov chain with state space $W$ and transition probability matrix $Q = [Q_{kl}]$. Furthermore, assume that $X$ and $Y$ are independent. Let $Z={Z_n}$ be the process $Z_n.= (X_n,Y_n)$. (a) Identify the state space of $Z$. (b) Assuming that $Z={Z_n}$ is a Markov chain, identify the transition probability matrix of $Z$.

For part a, I know that the state space is the possible values that the chain Z can take on. I'm having trouble understanding what the state space of the process $Z_n$ is. Is it ${S, W}$ because Z is now that of the process $(X_n, Y_n)$? Or is it all possible combinations of $(S, W)$ because $Z_n$ can be all combinations of $(X_n, Y_n)$, which exist as some value from $(S,W)$?

I'm not sure how to approach part b. I am given the transition probability matrices of $X$ and $Y$, but I'm not sure what their relation is to $Z$.

Answer 1

State space is $S\times W$ with transition matrix $Q\otimes P$ where this denotes the matrix tensor (kronecker) product (see wikipedia)

So if $S=\{s_1,\ldots,s_a\}$ and $W=\{w_1,\ldots,w_b\}$ are the state spaces, the new state space is $$\{(s_1,w_1),\ldots,(s_1,w_b),(s_2,w_1),\ldots,(s_2,w_b),\ldots,(s_a,w_1),\ldots,(s_a,w_b))$$