let $X_t = 0.5X_{t-1} + Z_t$ where $Z_t$ ~ $ WN(0,\sigma^2)$
I want to find the ACVF of both $X_t$ and $Z_t$, but I am a little bit confused. Say for $X_t$ $$\gamma(h) = COV(0.5X_{t-1} + Z_t, 0.5X_{t-1+h} + Z_{t+h}$$ $ = 0.5^2COV(X_{t-1},X_{t-1+h}) + 0.5COV(X_{t-1},Z_{t+h}) + 0.5COV(Z_{t},X_{t-1+h}) + COV(Z_{t},Z_{t+h}) $
then for say $h =-1$ could I still say that the $0.5^2 COV(X_{t-1},Z_{t+h}) = 0.5 \sigma^2$ ? Or is there a difference because of the differing $X $ and $Z$ terms
For every $t$, $$X_t=\sum_{n\geqslant0}a_nZ_{t-n},\qquad a_n=\text{____},$$ hence, for $h\geqslant0$, $$X_{t+h}X_t=\sum_{n,m\geqslant0}a_na_mZ_{t+h-m}Z_{t-n}.$$ By hypothesis, $E(Z_rZ_s)=0$ except when $r=s$ and then $E(Z_r^2)=\sigma^2$ hence $$E(X_{t+h}X_t)=\sigma^2\sum_{n,m\geqslant0}a_na_m\,[t+h-m=t-n]=\sigma^2\sum_{n\geqslant0}a_na_{h+n}.$$ Now the series on the right is easy to evaluate once $(a_n)$ has been identified.