A Complete Asymptotic Series for the Autocovariance Function of a Long Memory Process

An infinite-order asymptotic expansion is given for the autocovariance function of a general stationary long-memory process with memory parameter d in (-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/k1-2d), 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/k3-2d). The derivation uses Erdélyi’s (1956) expansion for Fourier-type integrals when there are critical points at the boundaries of the range of integration - here the frequencies {0,2}. 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.