mean square convergence of the sample mean of a stationary process
If {Xt,t∈T} is a stationary process with mean μ and autocovariance function γ(⋅), then as n→∞ we have the following:
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var[ˉXn]=E[(ˉXn-μ)2]→0 if γ(n)→0
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nE[(ˉXn-μ)2]→∑∞h=-∞γ(h) if ∑∞h=-∞|γ(h)|<∞ where
ˉXn=1nn∑k=1Xk is the sample mean
which is a natural unbiased estimator
of the mean μ of the stationary process {Xt}.
References
- 1 Peter J. Brockwell G., Richard A. Davis , Time Series :Theory and Methods.
Title | mean square convergence of the sample mean of a stationary process |
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Canonical name | MeanSquareConvergenceOfTheSampleMeanOfAStationaryProcess |
Date of creation | 2013-03-22 15:20:52 |
Last modified on | 2013-03-22 15:20:52 |
Owner | georgiosl (7242) |
Last modified by | georgiosl (7242) |
Numerical id | 5 |
Author | georgiosl (7242) |
Entry type | Theorem |
Classification | msc 60G10 |