Asymptotic theory for the detection of mixing in anomalous diffusion

TL;DR

This paper develops an asymptotic theory for detecting mixing in anomalous diffusion processes, enabling robust hypothesis testing based on single sample paths for a broad class of Gaussian processes.

Contribution

It provides the first comprehensive asymptotic analysis of a mixing detection method applicable to various Gaussian processes, including fractional Gaussian noise.

Findings

Asymptotic distribution can be Gaussian or non-Gaussian depending on the diffusion exponent.

Convergence rates vary, being standard or nonstandard based on process parameters.

Results facilitate mixing detection from a single observed sample path.

Abstract

In this paper, we develop asymptotic theory for the mixing detection methodology proposed by M. Magdziarz and A. Weron [Physical Review E, 84:051138 (2011)]. The assumptions cover a broad family of Gaussian stochastic processes including fractional Gaussian noise and the fractional Ornstein-Uhlenbeck process. We show that the asymptotic distribution and convergence rates of the detection statistic may be, respectively, Gaussian or non-Gaussian and standard or nonstandard depending on the diffusion exponent. The results pave the way for mixing detection based on a single observed sample path and by means of robust hypothesis testing.