Consider now the sample variance estimator
where the mean is assumed to be , and denotes the unbiased sample autocorrelation of based on the samples leading up to and including time . Since is unbiased, . The variance of this estimator is then given by
by the independence of and , and when , the fourth moment is given by . More generally, we can simply label the th moment of as , where corresponds to the mean, corresponds to the variance (when the mean is zero), etc.
When is assumed to be Gaussian white noise, we have
so that the variance of our estimator for the variance of Gaussian white noise is
Again we see that the variance of the estimator declines as .
The same basic analysis as above can be used to estimate the variance of the sample autocorrelation estimates for each lag, and/or the variance of the power spectral density estimate at each frequency.