mean square convergence of the sample mean of a stationary process
If $\{{X}_{t},t\in T\}$ is a stationary process with mean $\mu $ and autocovariance function $\gamma (\cdot )$, then as $n\to \mathrm{\infty}$ we have the following:

•
$\mathrm{var}[{\overline{X}}_{n}]=E[{({\overline{X}}_{n}\mu )}^{2}]\to 0$ if $\gamma (n)\to 0$

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$nE[{({\overline{X}}_{n}\mu )}^{2}]\to {\sum}_{h=\mathrm{\infty}}^{\mathrm{\infty}}\gamma (h)$ if $$ where
$${\overline{X}}_{n}=\frac{1}{n}\sum _{k=1}^{n}{X}_{k}$$ is the sample mean^{} which is a natural unbiased estimator^{} of the mean $\mu $ of the stationary process $\{{X}_{t}\}$.
References
 1 Peter J. Brockwell G., Richard A. Davis , Time Series $\mathrm{:}$Theory and Methods.
Title  mean square convergence of the sample mean of a stationary process 

Canonical name  MeanSquareConvergenceOfTheSampleMeanOfAStationaryProcess 
Date of creation  20130322 15:20:52 
Last modified on  20130322 15:20:52 
Owner  georgiosl (7242) 
Last modified by  georgiosl (7242) 
Numerical id  5 
Author  georgiosl (7242) 
Entry type  Theorem 
Classification  msc 60G10 