Testing of fractional Brownian motion in a noisy environment

TL;DR

This paper develops a statistical test based on autocorrelation functions to detect fractional Brownian motion in noisy environments, providing explicit distribution formulas and evaluating its effectiveness against existing methods.

Contribution

It introduces a new rigorous ACF-based test for FBM with white noise, deriving its distribution and demonstrating its applicability and power compared to prior tests.

Findings

Test statistic follows a generalized chi-squared distribution.

The test effectively distinguishes FBM in noisy data.

Comparison shows improved power over existing tests.

Abstract

Fractional Brownian motion (FBM) is the only Gaussian self-similar process with stationary increments. Its increment process, called fractional Gaussian noise, is ergodic and exhibits a property of power-like decaying autocorrelation function (ACF) which leads to the notion of long memory. These properties have made FBM important in modelling real-world data recorded in different experiments ranging from biology to telecommunication. These experiments are often disturbed by a noise which source can be just the instrument error. In this paper we propose a rigorous statistical test based on the ACF for FBM with added white Gaussian noise. To this end we derive a distribution of the test statistic which is given explicitly by the generalized chi-squared distribution. This allows us to find critical regions for the test with a given significance level. We check the quality of the introduced test by studying its power and comparing with other tests existing in the literature. We also note that the introduced test procedure can be applied to an arbitrary Gaussian process.