Asymptotic theory for the detection of mixing in anomalous diffusion

Abstract

In this paper, starting from the methodology proposed in Magdziarz and Weron (2011), we develop asymptotic theory for the detection of mixing in Gaussian anomalous diffusion. The assumptions cover a broad family of 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.