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

Fractional Gaussian noise and $\text{FARIMA}(0,d,0)$ have the same long-memory pole $\mathrm{\mid }\theta {\mathrm{\mid }}^{-2d}$ and hence the same leading PACF law $\alpha (n)\sim d\mathrm{/}n$. We show that this agreement breaks at the first non-universal order. For $0<d<1\mathrm{/}2$, the pure fGn PACF satisfies $${\alpha }_{\text{fGn}}(n)=\frac{d}{n}+\frac{C_{\text{fGn}}(d)}{n^2}+o(n^{-2}),C_{\text{fGn}}(d)<d^2,$$ 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.