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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)∼ d/n. We show that this agreement breaks at the first non-universal order. For 0<d<1/2, the pure fGn PACF satisfies αfGn(n)=d/n+CfGn(d)/n2+o(n−2), CfGn(d)<d2, 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.
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BibTeX
@misc{cai2026secondorder,
title = {{Second-order PACF asymptotics and discrimination between fractional Gaussian noise and $\FARIMA(0,d,0)$}},
author = {Chunhao Cai},
howpublished = {arXiv},
year = {2026},
month = nov,
doi = {10.48550/arxiv.2605.31416},
eprint = {2605.31416},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2605.31416},
}
RIS
TY - UNPB
TI - Second-order PACF asymptotics and discrimination between fractional Gaussian noise and FARIMA(0,d,0)
AU - Chunhao Cai
PY - 2026
DA - 2026/11/05
PB - arXiv
DO - 10.48550/arxiv.2605.31416
UR - https://arxiv.org/abs/2605.31416
AB - 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)∼ 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)=d/n+C_fGn(d)/n^2+o(n^−2), 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.
ER -