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 thefollowing:

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

• $nE[{({\overline{X}}_n-\mu )}^2]\to {\sum }_{h=-\mathrm{\infty }}^{\mathrm{\infty }}\gamma (h)$ ifwhere

  $${\overline{X}}_n=\frac{1}{n}\sum _{k=1}^n{X}_k$$

  is the sample mean which is a natural unbiased estimator of the mean $\mu$ of the stationary process$\{X_t\}$.