More limit theory for the sample correlation function of moving averages

Abstract

Let X t = Σ ∞ j=-∞ c j Z t - j be a moving average process where { Z t } is iid with common distribution in the domain of attraction of a stable law with index α, 0 < α < 2. If 0 < α < 2, E| Z 1| α < ∞ and the distribution of | Z 1|and | Z 1 Z 2| are tail equivalent then the sample correlation function of { X 1} suitably normalized converges in distribution to the ratio of two dependent stable random variables with indices α and α/2. This is in sharp contrast to the case E| Z 1| α = ∞ where the limit distribution is that of the ratio of two independent stable variables. Proofs rely heavily on point process techniques. We also consider the case when the sample correlations are asymptotically normal and extend slightly the classical result.