Second-order PACF asymptotics and discrimination between fractional Gaussian noise and $\FARIMA(0,d,0)$
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arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2026
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| _version_ | 1866918531708223488 |
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| author | Cai, Chunhao |
| author_facet | Cai, Chunhao |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2605_31416 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Second-order PACF asymptotics and discrimination between fractional Gaussian noise and $\FARIMA(0,d,0)$ Cai, Chunhao Statistics Theory Probability 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. |
| title | Second-order PACF asymptotics and discrimination between fractional Gaussian noise and $\FARIMA(0,d,0)$ |
| topic | Statistics Theory Probability |
| url | https://arxiv.org/abs/2605.31416 |