## Autocorrelation of a random process

Hello Everyone!

I got this question:

X[n] and Y[n] are two DT independent DT processes, and Z[n] = X[n] x Y[n]. If X[n] and Y[n] are JWSS, is Z[n] WSS?

Here's what I'm doing

$R_Z(m,l)=E[X[m]Y[l]X[m]Y[l]]=\sum _i \sum_j W_i Z_i P(W,Z)$

$=\sum _i \sum_j X_i Y_i X_j Y_j P(W,Z)$.

Now, this is what I know:

$P(X,Y) = P(X) \times P(Y)$ and $R_{XY}[m,l]=\sum _i \sum_j X_i Y_j P(X,Y)$ is a function of $m-l$.

Now, I'm stuck because I can't "expand" the probability $P(W,Z)$. So I ask this:

If we have a process X[n], $R_X = \sum _i \sum _j X_i X_j P(X_i, X_j)$, is $P(X_i, X_j) = P(X_i) \times P(X_j) = P(X) \times P(X)$ ?

Note: If the expressions are written correctly please point that out.