Fractional Gaussian noise: Projections, prediction, norms

TL;DR

This paper analyzes projections of fractional Gaussian noise onto neighboring elements, deriving analytical results and recurrence relations, and studying how these projections behave with respect to the Hurst index and projection size.

Contribution

It provides new analytical formulas, recurrence relations, and numerical insights into the projections and norms of fractional Gaussian noise.

Findings

Projection coefficients depend on the Hurst index and number of neighbors

Recurrence relations for two-sided projection coefficients are established

Norms of projections are characterized for various cases

Abstract

We examine the one-sided and two-sided (bilateral) projections of an element of fractional Gaussian noise onto its neighboring elements. We establish several analytical results and conduct a numerical study to analyze the behavior of the coefficients of these projections as functions of the Hurst index and the number of neighboring elements used for the projection. We derive recurrence relations for the coefficients of the two-sided projection. Additionally, we explore the norms of both types of projections. Certain special cases are investigated in greater detail, both theoretically and numerically.