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

Hosking, J. R. M. (1996). Asymptotic distributions of the sample mean, autocovariances and autocorrelations of long-memory time series. Journal of Econometrics, 73, 261-284.

Abstract. We derive the asymptotic distributions of the sample mean, autocovariances and autocorrelations for a time series whose autocovariance function gamma sub k

has the power-law decay
gamma sub k like lambda k sup -alpha

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 < alpha <= 1/2

are not of asymptotic order n^-1. When 0 < alpha < 1/2

the asymptotic distributions of the sample autocovariances and autocorrelations are not Normal.

Author's note. Supplementary results, mostly details of proofs, are available in IBM Research Report RC19831.

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