Some robust exact results on sample autocorrelations and tests of randomness

Several exact results on the second moments of sample autocorrelations, for both Gaussian and non-Gaussian series, are presented. General formulae for the means, variances and covariances of sample autocorrelations are given for the case where the variables in a sequence are exchangeable. Bounds for the variances and covariances of sample autocorrelations from an arbitrary random sequence are derived. Exact and explicit formulae for the variances and covariances of sample autocorrelations from a Gaussian white noise are given. It is observed that the latter results hold for all spherically symmetric distributions. A simulation experiment, with Gaussian series, indicates that normalizing each sample autocorrelation with its exact mean and variance, instead of the usual approximate moments, can improve considerably the accuracy of the asymptotic N(0,1) distribution to obtain critical values for tests of randomness. The exact second moments of rank autocorrelations are also studied.