Evaluate the mean and covariance function for each of the following processes. In each case, determine whether or not the process is stationary.
a) $Y_t$ = $\theta_0$ +t$e_t$
b) $W_t$ = $\nabla$$Y_t$ where $Y_t$ is given in part a).
c) $Y_t$=$e_t$*$e_{t-1}$
for part a)
E[$Y_t$]=E[$\theta_0$ +t$e_t$] = $\theta_0$+tE[$e_t$]=$\theta_0$ (due to 0 mean)
then Covariance = E[$Y_t$$Y_s$]-E[$Y_t$]E[$Y_s$]
where E[$Y_t$$Y_s$]=E[($\theta_0$ +t$e_t$)($\theta_0$ +s$e_s$)]=$\theta_0^2$+$\theta_0$sE[$e_s$]+$\theta_0$tE[$e_t$]+$ts^2$ E[$e_t$$e_s$] = $\theta_0^2$
This would mean that Covariance = $\theta_0^2$ - $\theta_0$$\theta_0$=0??