$\newcommand{\cov}{{\rm\mathbb{C}ov}}$ My question is about the final steps in the calculation of the covariance of the AR(1) Model. I Don't see how the Power K can be replaced by H+J.
\begin{eqnarray} \cov(X_t,X_{t + h}) &=& \cov\left(\sum_{k=0}^{\infty}\phi^k\varepsilon_{t-k},\sum_{j=0}^{\infty}\phi^j\varepsilon_{t-h-j}\right) \\ &=&\sum_{k=0}^{\infty}\sum_{j=0}^{\infty}\phi^k\phi^j\cov(\varepsilon_{t-k}, \varepsilon_{t-h-j}) =\sum_{j=0}^{\infty}\phi^{h+j}\phi^j\sigma_{\varepsilon}^2\\ &=&\sum_{j=0}^{\infty}\phi^{h}\phi^{2j}\sigma_{\varepsilon}^2 = \phi^{h}\sum_{j=0}^{\infty}\phi^{2j}\sigma_{\varepsilon}^2 = \phi^h\gamma(0) \end{eqnarray}
Because the $\epsilon$ are uncorrelated so all the covariance's are zero except when $t-k=t-h-j$