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

Abstract

We derive the asymptotic distributions of the sample mean, autocovariances, and autocorrelations for a time series whose autocovariance function {γk} has the powerlaw decay γk ∼ λk-α, λ > 0, 0 < α < 1, as k → ∞. 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 ≤ 1/2, are not of asymptotic order n-1. When 0 < α < 1/2 the asymptotic distributions of the sample autocovariances and autocorrelations are not Normal.