Second-order PACF asymptotics and discrimination between fractional Gaussian noise and $\FARIMA(0,d,0)$

Abstract

Fractional Gaussian noise and $\FARIMA(0,d,0)$ have the same long-memory pole $|θ|^{-2d}$ and hence the same leading PACF law $α(n)\sim d/n$. We show that this agreement breaks at the first non-universal order. For $0<d<1/2$, the pure fGn PACF satisfies $$ α_{\fGn}(n)=\frac d n+\frac{C_{\fGn}(d)}{n^2}+o(n^{-2}), \qquad C_{\fGn}(d)<d^2, $$ The proof uses the Bingham--Inoue--Kasahara representation, a phase-coefficient expansion for fGn, and a Hankel-operator perturbation argument. Thus the fGn spectral envelope is invisible at first order but visible in second-order finite prediction, explaining why short-memory order selection can differ when fGn data are fitted by FARIMA-type models.