A complete asymptotic series for the autocovariance function of a long memory process

Abstract

Abstract An infinite-order asymptotic expansion is given for the autocovariance function of a general stationary long-memory process with memory parameter d ∈ ( − 1 / 2 , 1 / 2 ) . The class of spectral densities considered includes as a special case the stationary and invertible ARFIMA( p , d , q ) model. The leading term of the expansion is of the order O ( 1 / k 1 − 2 d ) , where k is the autocovariance order, consistent with the well known power law decay for such processes, and is shown to be accurate to an error of O ( 1 / k 3 − 2 d ) . The derivation uses Erdelyi’s [Erdelyi, A., 1956. Asymptotic Expansions. Dover Publications, Inc, New York] expansion for Fourier-type integrals when there are critical points at the boundaries of the range of integration - here the frequencies { 0 , 2 π } . Numerical evaluations show that the expansion is accurate even for small k in cases where the autocovariance sequence decays monotonically, and in other cases for moderate to large k . The approximations are easy to compute across a variety of parameter values and models.