Assume $X_n$ is an iid gaussian random process with zero mean and variance $\sigma^2$, and $U_n$ be an iid binary random process with $P_r\{ U_{n}=1\}=P_r\{U_n=-1\}=0.5$, and $\{U_n\}$ is independent of $\{X_n\}$, now let $Z_n=X_n U_n$.
Now,i want to prove the $Z_n$ is WSS, and we know if some random process is WSS, it should satisfy these two properties
$1$. Mean is a constant value
$2$. Autocorrelation is just related to time difference
And i know the mean of $Z$ is zero, and zero is a constant, but how do i prove the second property?
It's easy to see that $Z_n$ has the same distribution as $X_n$, but I leave the argumentation to you (does an equiprobable sign flip change a symmetric PDF?). The autocorrelation of $Z_n$ is thus the same as that of $X_n$, i.e. \begin{align*} R_{ZZ}(m,n) &= R_{XX}(m,n) = \mathrm{E}[X_m X_n] = \sigma^2 \delta_{m,n} \\ &= \left\{\begin{array}{ll} \sigma^2 & \text{if } m = n \\ 0 & \text{otherwise} \end{array}\right. \\ &= \left\{\begin{array}{ll} \sigma^2 & \text{if } m-n = 0 \\ 0 & \text{if } m-n \neq 0 \end{array}\right. \\ &= \tilde{R}_{ZZ}(m-n) \end{align*} which is a function of $m-n$.