Analytical and computational problems related to fractional Gaussian noise

TL;DR

This paper investigates the projection properties of fractional Gaussian noise, deriving recurrence relations, conjecturing behaviors, and analyzing the covariance function's properties, with implications for understanding its structure and behavior.

Contribution

It provides new analytic results, recurrence relations, and conjectures about the coefficients of projections in fractional Gaussian noise, along with properties of its covariance function.

Findings

Recurrence relations for projection coefficients

Conjectures supported by numerical evidence

Covariance function is completely monotone for H>1/2

Abstract

We study the projection of an element of fractional Gaussian noise onto its neighbouring elements. We prove some analytic results for the coefficients of this projection, in particular, we obtain recurrence relations for them. We also make several conjectures concerning the behaviour of these coefficients, provide numerical evidence supporting these conjectures, and study them theoretically in particular cases. As an auxiliary result of independent interest, we investigate the covariance function of fractional Gaussian noise, prove that it is completely monotone for $H>1/2$, and, in particular monotone, convex, log-convex along with further useful properties.