Fractional Brownian Motion Approximation Based on Fractional Integration of a White Noise

TL;DR

This paper introduces simple spectral-based approximations for fractional Gaussian noise and fractional Brownian motion using fractional integration of white noise, highlighting their correlation, self-similarity, and potential for modeling natural processes.

Contribution

It proposes novel spectral approximation methods for fractional Gaussian noise and fractional Brownian motion based on fractional integration of white noise.

Findings

The approximations capture persistent and anti-persistent behaviors.

They exhibit self-similar properties consistent with theoretical laws.

The models are useful for natural process modeling and data analysis.

Abstract

We study simple approximations to fractional Gaussian noise and fractional Brownian motion. The approximations are based on spectral properties of the noise. They allow one to consider the noise as the result of fractional integration/differentiation of a white Gaussian noise. We study correlation properties of the approximation to fractional Gaussian noise and point to the peculiarities of persistent and anti-persistent behaviors. We also investigate self-similar properties of the approximation to fractional Brownian motion, namely, ``tH laws`` for the structure function and the range. We conclude that the models proposed serve as a convenient tool for the natural processes modelling and testing and improvement of the methods aimed at analysis and interpretation of experimental data.