Let be $\{w_t\}_{t\geq0}$ white noise with zero mean and variance $\sigma^2$. Define the process $X(t)=w_tw_{t-a}$. Is $X(t)$ a wide-sense stationary process?
I'm trying to prove that $E[X(t)]$ is time independent and the correlation $R(t,t+k)$ only depends of $k$.
$$E[X(t)]=E[w_tw_{t-a}]=E[w_tw_{t-a}]-E[w_t]E[w_{t-a}]=Cov(w_t,w_{t-a})=0$$ so, the mean function of the process $X(t)$ is independent of the time $t$.
$$R(t,t+k)=\frac{E[(X_{t}-E[X(t)])(X_{t+k}-E[X(t+k))]}{\sigma_t\sigma_{t+k}}$$ $$R(t,t+k)=\frac{E[(X_{t}-0)(X_{t+k}-0)]}{\sigma_t\sigma_{t+k}}=\frac{E[X_{t}X_{t+k}]}{\sigma_t\sigma_{t+k}}=\frac{E[w_tw_{t-a}w_{t+k}w_{t+k-a}]}{\sigma_t\sigma_{t+k}}. $$ $$\sigma_t=\sqrt{E[X(t)^2]-E^2[X(t)]}=\sqrt{E[w_t^2w_{t-a}^2]}$$
how can I prove that $R(t,t+k)$ only depends of $k$?
