# proof of mean square convergence of the sample mean of a stationary process

 $n\operatorname{var}(\bar{X}_{n})=\frac{1}{n}\sum_{i=1}^{n}\sum_{j=1}^{n}% \operatorname{cov}(X_{i},X_{j})\\ =\sum_{|h|

If $\gamma(n)\to 0$ as $n\to\infty$ then $\lim_{n\to\infty}\frac{1}{n}\sum_{|h|, whence $\operatorname{var}[\bar{X}_{n}]\to 0$.
If $\sum_{h=-\infty}^{\infty}|\gamma(h)|<\infty$ then the dominated Convergence theorem gives

 $\lim_{n\to\infty}\sum_{|h|

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Title proof of mean square convergence of the sample mean of a stationary process ProofOfMeanSquareConvergenceOfTheSampleMeanOfAStationaryProcess 2013-03-22 15:22:19 2013-03-22 15:22:19 georgiosl (7242) georgiosl (7242) 6 georgiosl (7242) Proof msc 60G10