Finding System transfer function, $H(z)$ or, equivalently, its impulse response, $h[n]$
From only the process statistics, you cannot derive the exact impulse response/transfer function. However, by the Wiener-Khinchin-Theorem you know, that the power spectrum density (PSD) of an input signal to an LTI system is transformed with the magnitude-square of the transfer function:

Let $S_x(\omega)$ be the PSD of the input and $S_y(\omega)$ be the PSD at the output of the system. Note that 

$$\begin{align}
 S_x(\omega) &amp;= \mathrm{DTFT} \{ R_x[n] \} \\
             &amp;= |X(e^{j\omega})|^2 \\
\end{align}$$

where $X(e^{j\omega})= \mathrm{DTFT} \{x[n]\}$ and similarly for $S_y(\omega)$.

Then, you have

$$
S_y(\omega)=|H(e^{j\omega})|^2 S_x(\omega)
$$

and you can at least calculate the magnitude of $H(e^{j\omega})$ by inverting the equation.