The Properties of Fractional Gaussian Process and Their Applications

TL;DR

This paper explores fractional Gaussian processes, their mathematical properties, and applications, including statistical estimation bounds and connections to fractional Brownian motion, expanding understanding of non-stationary Gaussian processes.

Contribution

It characterizes fractional Gaussian processes, relates their RKHS to fractional Brownian motion, and extends applications like Berry-Esséen bounds for ergodic fractional Ornstein-Uhlenbeck processes.

Findings

Reproducing kernel Hilbert space of fractional Gaussian processes related to that of fractional Brownian motion.

Berry-Esséen bounds established for statistical estimation of fractional Ornstein-Uhlenbeck processes.

Identification of seven Gaussian processes with non-stationary increments within this framework.

Abstract

The process $(G_t)_{t\in[0,T]}$ is referred to as a fractional Gaussian process if the first-order partial derivative of the difference between its covariance function and that of the fractional Brownian motion $(B^H_t)_{t\in[0,T ]}$ is a normalized bounded variation function. We quantify the relation between the associated reproducing kernel Hilbert space of $(G)$ and that of $(B^H)$. Seven types of Gaussian processes with non-stationary increments in the literature belong to it. In the context of applications, we demonstrate that the Gladyshev's theorem holds for this process, and we provide Berry-Ess\'{e}en upper bounds associated with the statistical estimations of the ergodic fractional Ornstein-Uhlenbeck process driven by it. The second application partially builds upon the idea introduced in \cite{BBES 23}, where they assume that $(G)$ has stationary increments. Additionally, we briefly discuss a variant of this process where the covariance structure is not entirely linked to that of the fractional Brownian motion.