On the asymptotic expansion of the emphirical process of long-memory moving averages

Abstract

Let Xn = Σ∞i = 1αiεn-i, where the εi are iid with mean 0 and finite fourth moment and the αi are regularly varying with index - β where β ∈ (1/2, 1) so that {Xn} has long-range dependence. This covers an important class of the fractional ARIMA process. For r ≥ 0, let YN, r = ΣNn = 1Σ1 ≤ j1 < jrΠrs = 1αjsεn - js, YN , 0 = N, σ2N, r = Var(YN, r) and F(r) = the rth derivative of the distribution function of Xn. The YN, r are uncorrelated and are stochastically decreasing in r. For any positive integer p < (2β - 1)-1, it is shown under mild regularity conditions that, with probability 1, ΣN I(Xn ≤ x) = ΣP;(- 1)rF(r)(x)YN, r + o(N-λ σN, p) n = 1 uniformly for all x ∈ ℜ ∀ 0 < λ