= Autocovariance of an AR(1) process
{title2=$\gamma(h)=\sigma^2\phi^{|h|}/(1-\phi^2)$}
For the stationary causal <AR(1)> with $|\phi|<1$, matching terms in its <infinite moving-average representation> gives, for $h\ge0$,
$$
\gamma(h)=\sigma^2\sum_{j=0}^{\infty}\phi^j\phi^{j+h}
=\frac{\sigma^2\phi^h}{1-\phi^2}.
$$
Symmetry of the <autocovariance> extends this to negative lags. Dividing by $\gamma(0)$ gives the <autocorrelation> $\rho(h)=\phi^{|h|}$. The <geometric series> explains both the enlarged stationary <variance> and the diminishing memory: all past disturbances contribute to the present, but their squared weights have a finite sum. At $\phi=0$, the process reduces to shifted <white noise> and all nonzero-lag <autocorrelations> vanish. See https://www.stat.berkeley.edu/~bartlett/courses/153-fall2010/lectures/4.pdf[Bartlett, lecture 4: AR(1) representation and mean-square convergence].
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