If $X_t$ and $Y_t$ are independent and both are second order stationary processes, is $X_tY_t$ also stationary?
I need to show that
i) $E(X_tY_t)$ is time independent
ii) $Var(X_tY_t)<\infty$
iii)$Cov(X_{t+h}Y_{t+h},X_tY_t)$ only depends of $h$
Then
i) $E(X_tY_t)=\mu_x\mu_y$ since $X_t$ and $Y_t$ are stationary
ii)$Var(X_tY_t)=Var(X_t)Var(Y_t)+Var(X_t)E(Y_t)^2+Var(Y_t)E(X_t)^2<\infty$ since each term in the sum is finite
iii) $\gamma_{XY}(h)=Cov(X_{t+h}Y_{t+h},X_tY_t)=E(X_{t+h}Y_{t+h}X_tY_t)-E(X_{t+h}Y_{t+h})E(X_tY_t)$
since the processes are independent $$\gamma_{XY}(h)=[\gamma_X(h)+E(X_{t+h})E(X_t)][\gamma_Y(h)+E(Y_{t+h})E(Y_t)]-E(X_{t+h}Y_{t+h})E(X_tY_t)$$
$$=\gamma_X(h)\gamma_Y(h)+\gamma_X(h)E(Y_{t+h})E(Y_t)+\gamma_Y(h)E(X_{t+h})E(X_t)$$
I did something wrong? How I can proceed with this covariance?