I wonder if the autoregressive process of order 1 is an autocovariance-ergodic process (https://en.wikipedia.org/wiki/Ergodic_process): hence, if the time average estimate of autocovariance converges in squared mean to the ensemble average. In practice I have $$ Z(t+1) = aZ(t) + \sqrt{1-a^2}U(t+1) $$ with $\mathbb{E}[U(t)] = 0$, $var(U(t))=1$, $a\in (0,1)$, and $U(t)$ are i.i.d.. Setting $Z(1) = U(1)$ we obtain that $\mu_Z = \mathbb{E}[Z(t)] = 0$ and $var(Z(t)) = 1$. By induction, it can be shown that the first moment (expected value) is equal to $0$ and the second moment (autocovariance) is equal to $C(h) = a^h$ where $h$ is the lag. Hence, the process is stationary of order two. My question is if the sample autocovariance converges to the theoretical autocovariance in squared mean as the number of samples goes to $\infty$. I recall that the sample autocovariance, in this case, should be almost equal to the autocorrelation since the mean of the process is $0$. The sample estimate of the autocovariance is: $$\widehat{C}(h) = \dfrac{1}{N-h-1}\sum_{t=0}^{N-h-1} (Z(t+h)-\mu_Z)(Z(t)-\mu_Z).$$ I reiterate that I should verify this limit: $$ \lim_{N \rightarrow \infty} \mathbb{E}[(C(h) - \widehat{C}(h))^2] = 0$$


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