Asymptotic analysis of the finite predictor for the fractional Gaussian noise
P. Chigansky, M. Kleptsyna
TL;DR
This paper develops a Hilbert boundary value problem framework to obtain exact asymptotics for the finite predictor of stationary ARIMA-type processes driven by fractional Gaussian noise with long-range dependence. By constructing a sectionally holomorphic extension $Q(z)$ of the driving spectral density and translating the predictor equations into a Hilbert problem, the authors derive closed-form asymptotics for the relative prediction error and partial correlations, notably $\\delta(n) \,\sim \, \sigma^2_0 d^2 / n$ and $\alpha(n) \,\sim \, d/n$ for memory parameter $d\in(-1/2,1/2)\setminus\{0\}$. The approach unifies the treatment of long-memory behavior and MA polynomial zeros, yielding universal leading terms that depend only on $d$ (and on zeros when present) and providing explicit constants via the Szegö–Kolmogorov framework adjusted for sectionally holomorphic extensions. The results advance understanding of finite-predictor asymptotics in long-memory ARIMA models, with potential impact on time-series forecasting and spectral-domain prediction theory. The methodology leverages polylogarithmic representations, Vandermonde-based asymptotics, and boundary-value problem techniques to achieve exact, model-specific asymptotics beyond prior geometric-rate results.
Abstract
The goal of this paper is to propose a new approach to asymptotic analysis of the finite predictor for stationary sequences. It produces the exact asymptotics of the relative prediction error and the partial correlation coefficients. The assumptions are analytic in nature and applicable to processes with long range dependence. The ARIMA type process driven by the fractional Gaussian noise (fGn), which previously remained elusive, serves as our study case.