Asymptotic theory for the detection of mixing in anomalous diffusion

Abstract

In this paper, we develop asymptotic theory for the mixing detection methodology proposed by Magdziarz and Weron [Phys. Rev. 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.