### Estimating Autocorrelations If we assume a stationary time series is an instance of a stochastic process, then the [[Autocorrelations|ac.f. and acv.f. of the time series]] is an estimator of the [[Stochastic Processes#Describing Stochastic Processes|ac.f. and acv.f. of the stochastic process]]. The estimate is biased with $O(N^{-1})$ bias and consistent. If $(Z_{t})$ is a [[Purely Random Processes|PRP]], then the autocorrelation $r_{k}$ of a particular realization $(z_{t})$ has $\begin{align*} \mathbb{E}[r_{k}]&\approx -N^{-1}\\ \mathrm{Var}(r_{k}) &\approx N^{-1} \end{align*}$and asymptotically normal under some weak assumptions. This provides a baseline for the magnitude of autocorrelations. ### Estimating Means The sample mean $\bar{X}$ of a time series is only useful if we assume its underlying stochastic process is stationary. When the time series are not independent, the mean has variance $\mathrm{Var}(\bar{X})=\frac{\sigma^{2}}{N}\left[ 1+\sum_{r=1}^{N-1}\left( 1-\frac{r}{N} \right)\rho(r) \right]$where the factor to the right reduces the "effective" sample size if it is gt;0$.