I'm a little unsure if we have $(X_t)$ and $(Y_t)$ are two independent and strictly stationary processes would $(Z_t)$ given by $Z_t = X_tY_t$ also be strictly stationary?
I have had no problem showing it in case of stationarity, and i feel like it should be true, but I can't really figure out any sort of proof.
Yes, because $$(Z_{t_1 + h}, Z_{t_2 + h}, \ldots, Z_{t_n + h}) = (X_{t_1 + h}Y_{t_1 + h}, X_{t_2 + h}Y_{t_2 + h}, \ldots, X_{t_n + h}Y_{t_n + h})$$ $$ = f(X_{t_1 + h}, X_{t_2 + h}, \ldots, X_{t_n + h},Y_{t_1 + h}, Y_{t_2 + h}, \ldots, Y_{t_n + h}) $$ where $f(a_1, a_2, \ldots, a_n, b_1, b_2, \ldots, b_n) = (a_1b_1, a_2b_2, \ldots, a_nb_n)$. We have $(X_{t_1 + h}, X_{t_2 + h}, \ldots, X_{t_n + h}) \stackrel{d}{=} (X_{t_1 }, X_{t_2 }, \ldots, X_{t_n })$ and $(Y_{t_1 + h}, Y_{t_2 + h}, \ldots, Y_{t_n + h}) \stackrel{d}{=} (Y_{t_1 }, Y_{t_2 }, \ldots, Y_{t_n })$. Hence $(X_{t_1 + h}, X_{t_2 + h}, \ldots, X_{t_n + h},Y_{t_1 + h}, Y_{t_2 + h}, \ldots, Y_{t_n + h}) \stackrel{d}{=} (X_{t_1 }, X_{t_2 }, \ldots, X_{t_n }, Y_{t_1 }, Y_{t_2 }, \ldots, Y_{t_n })$ by independence.
Thus $$(Z_{t_1 + h}, Z_{t_2 + h}, \ldots, Z_{t_n + h}) = f(X_{t_1 + h}, X_{t_2 + h}, \ldots, X_{t_n + h},Y_{t_1 + h}, Y_{t_2 + h}, \ldots, Y_{t_n + h}) $$ $$ \stackrel{d}{=} f(X_{t_1 }, X_{t_2 }, \ldots, X_{t_n }, Y_{t_1 }, Y_{t_2 }, \ldots, Y_{t_n }) = (X_{t_1 }Y_{t_1 }, X_{t_2 }Y_{t_2 }, \ldots, X_{t_n }Y_{t_n})$$ $$=(Z_{t_1}, Z_{t_2}, \ldots, Z_{t_n}) .$$