Asymptotic distributions of the sample mean, autocovariances, and autocorrelations of long-memory time series

We derive the asymptotic distributions of the sample mean, autocovariances, and autocorrelations for a time series whose autocovariance function “γκ” has the powerlaw decay γκ ∼ λκ−Σ, λ > 0, o < α < 1, as κ → ∞. The results differ in important respects from the corresponding results for short-memory processes, whose autocovariance functions are absolutely summable. For long-memory processes the variances of the sample mean, and of the sample autocovariances and autocorrelations for 0 < α ⩽ 12, are not of asymptotic order n−1. When 0 < α < 12 the asymptotic distributions of the sample autocovariances and autocorrelations are not Normal.