Find process correlation function

Question

I have following task:
There are two normal centered random processes $X(t)$ and $Y(t)$ given. They have correlation functions $K_{x}(t_{1}, t_{2}), K_{y}(t_{1}, t_{2})$ and cross-correlation function $R_{xy}(t_{1}, t_{2})$. Get correlation function of process $Z(t) = X(t)Y(t)$.

What did I do:
$\bar Z = M[XY]$
$K_{Z} = M[(X(t_{1})Y(t_{1}) - M[X(t_{1})Y(t_{1})])\cdot (X(t_{2})Y(t_{2})-M[X(t_{2})Y(t_2)])] = M[X(t_1)X(t_2)Y(t_1)Y(t_2)]-M[X(t_1)Y(t_1)M[X(t_2)Y(t_2)]]-M[X(t_2)Y(t_2)M[X(t_1)Y(t_1)]]+M[M[X(t_1)Y(t_1)]M[X(t_2)Y(t_2)]] = M[X(t_1)X(t_2)Y(t_1)Y(t_2)] + M[X_1Y_1]M[X_2Y_2]$

But the answer is:
$K_x(t_1, t_2)\cdot K_y(t_1, t_2) + R_{xy}(t_1, t_2)R_{xy}(t_2, t_1) = M[X(t_1)X(t_2)]\cdot M[Y(t_1)Y(t_2)] + M[X(t_1)Y(t_2)]\cdot M[X(t_2)Y(t_1)]$

Honestly I don't know how to reduce my ideas to correct answer. Any ideas?

Answer 1

To calculate the correlation function of $ Z $, the following formula for 4th order moment of centered Gaussian random variables is useful. If $ (X_1,X_2,X_3,X_4) $ is a centered Gaussian 4d-random variable, then \begin{align*} \mathsf{M}[X_1X_2X_3X_4]&=\mathsf{M}[X_1X_2]\mathsf{M}[X_3X_4]\\ &\quad +\mathsf{M}[X_1X_3]\mathsf{M}[X_2X_4]+\mathsf{M}[X_1X_4]\mathsf{M}[X_2X_3].\tag{1} \end{align*} (cf. E. Parzen, Stochastic Processes, Holden-Day, 1962. p.92, Ch3, Th4C. ) Using (1), \begin{align*} &\mathsf{M}[X(t_1)X(t_2)Y(t_1)Y(t_2)] =\mathsf{M}[X(t_1)X(t_2)]\mathsf{M}[Y(t_1)Y(t_2)]\\ &\quad +\mathsf{M}[X(t_1)Y(t_1)]\mathsf{M}[X(t_2)Y(t_2)] +\mathsf{M}[X(t_1)Y(t_2)]\mathsf{M}[X(t_2)Y(t_1)]. \end{align*} Hence \begin{align*} K_Z(t_1,t_2)&= \mathsf{M}[X(t_1)Y(t_1)X(t_2)Y(t_2)]-\mathsf{M}[X(t_1)Y(t_1)]\mathsf{M}[X(t_2)Y(t_2)]\\ &=\mathsf{M}[X(t_1)X(t_2)]\mathsf{M}[Y(t_1)Y(t_2)] + \mathsf{M}[X(t_1)Y(t_2)]\mathsf{M}[X(t_2)Y(t_1)]\\ &=K_X(t_1,t_2) K_Y(t_1,t_2)+R_{XY}(t_1,t_2)R_{XY}(t_2,t_1). \end{align*}