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

Let Xₙ = Σᵢ₌₁∞ aᵢ εₙ₋ᵢ, where the εᵢ are iid with mean 0 finite fourth moment and the aᵢ are regularly varying with index -β where β ε (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 = ∑ₙ₌₁N ∑1≤ j₁ < < jᵣ Πₛ₌₁ʳ ajₛ, YN, 0 = N, σN, r² = (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, ∑ₙ₌₁N I(Xₙ ≤ x) = ∑ᵣ₌₀ᵖ (-1)ʳ F⁽ʳ⁾ (x) YN,r + o(N-λ σN,p)\\ uniformly for allx ε ∀ 0 < λ < (β - 1/2) (1/2 - p(β - 1/2)). This generalizes a host of existing results and provides the vehicle for a number of statistical applications.