# 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 $\gamma(\cdotp)$, then as $n\to\infty$ we have the following:

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

• $nE[(\bar{X}_{n}-\mu)^{2}]\to\sum_{h=-\infty}^{\infty}\gamma(h)$ if $\sum_{h=-\infty}^{\infty}|\gamma(h)|<\infty$ where

 $\bar{X}_{n}=\frac{1}{n}\sum_{k=1}^{n}X_{k}$

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 $\colon$Theory and Methods.
Title mean square convergence of the sample mean of a stationary process MeanSquareConvergenceOfTheSampleMeanOfAStationaryProcess 2013-03-22 15:20:52 2013-03-22 15:20:52 georgiosl (7242) georgiosl (7242) 5 georgiosl (7242) Theorem msc 60G10