Suppose I have two series:

$\displaystyle x_t=0.3x_{t-1}+w_t$

$\displaystyle y_t=x_t+0.5w_t$

where $\displaystyle w_t\sim WN(0,1)$.

I try to find the autocovariance function of $\displaystyle y_t$ as followed:

$\displaystyle \gamma(h)=Cov(y_{t-h},y_t)=Cov(x_{t-h}+0.5w_{t-h},x_t+0.5w_t)$

$\displaystyle =Cov(x_{t-h},x_t)+0.5Cov(w_{t-h},x_t)+0.5Cov(w_t,x_{t-h})+0.5^2Cov(w_{t-h},w_t)$

Then $\displaystyle \gamma(0)=1/(1-0.3^2)+0.5^2$

$\displaystyle \gamma(1)=0.3/(1-0.3^2)$

$\displaystyle \gamma(2)=0.3^2/(1-0.3^2)$ and so on..

My question is: am I right to assume both $\displaystyle 0.5Cov(w_{t-h},x_t)$ and $\displaystyle 0.5Cov(w_t,x_{t-h})$ are 0 at any lag h?

By the way, is the series $\displaystyle y_t$ weakly stationary such that $\displaystyle \gamma(h)$ is independent of t?