ASYMPTOTIC DISTRIBUTIONS OF THE SAMPLE-MEAN, AUTOCOVARIANCES, AND AUTOCORRELATIONS OF LONG-MEMORY TIME-SERIES

We derive the asymptotic distributions of the sample mean, autocovaria nces, and autocorrelations for a time series whose autocovariance func tion {gamma(k)} has the powerlaw decay gamma(k) similar to gamma k(-al pha), lambda > 0, 0 < alpha < 1, as k --> infinity. The results differ in important respects From the corresponding results for short-memory processes, whose autocovariance functions are absolutely summable. Fo r long-memory processes the variances of the sample mean, and of the s ample autocovariances and autocorrelations for 0 < alpha less than or equal to 1/2, are not of asymptotic order n(-1). When 0 < alpha < 1/2 the asymptotic distributions of the sample autocovariances and autocor relations are not Normal.