The Distribution and Quantiles of Sample Autocovariances and Autocorrelations of Sample Moments from a Stationary Process

This paper gives expansions for the distribution, density and quantiles of any estimate that is a smooth function of the sample cross-moments of a stationary process. Three versions of these are given, depending on whether an exact, approximate, or asymptotic form is used for the variance or covariance of the estimate. Eight examples are provided, including sample autocovariances and autocorrelations. Their Central Limit Theorems extend those in the literature, such as Bartlett’s formula, by allowing for the effect of the mean and higher order cross-cumulants. Their distribution and quantiles are given to magnitude $n^{-r/2}$ up to r = 3, where n is the sample size.