The variance of an increasingly smoothed random variable

The rate of decrease of the variance of a stationary random function with increasing smoothing is examined from the viewpoint of the definition of variance. Assuming the smoothing interval, t, to be small with respect to the length of record, and the variance to be calculated from overlapping running means, it is shown that the variance, σ2s, is related to smoothing interval by dσ2s/σ22,o = 2 [P(t/2)/σ2s -1] dt/t where P(t/2) is the autocovariance at lag t/2. Semi-empirical forms for P(t/2) are shown to provide a good fit for results from time series of both Gaussian and non-Gaussian distributions, and to yield fair estimates of the autocovariance as a function of lag time. (author)