Fractional Brownian motion (FBM) stands as a fundamental model for depicting dynamics in diverse complex systems. It is characterized by the Hurst exponent, influencing the correlation between FBM increments, its self-similarity, and anomalous diffusion behavior. Recent studies, however, suggest that the conventional model may fall short in describing experimental observations when the anomalous diffusion exponent varies across trajectories. Consequently, modifications to the classical FBM have been explored, including the extension to FBM with a random Hurst exponent.
We address the challenge of distinguishing between two models: (i) FBM with a constant Hurst exponent and (ii) FBM with a random Hurst exponent. The analysis focuses on the probabilistic properties of statistics represented by quadratic forms, which have found recent application in Gaussian processes and proven efficient for hypothesis testing. Two statistics—the sample autocovariance function and the empirical anomaly measure—are examined, leveraging the correlation properties of the considered models. A testing procedure is introduced based on these statistics to differentiate between the two models. Analytical and simulation results are presented, considering two-point and beta distributions as exemplary distributions of the random Hurst exponent.
To underscore the efficacy of the presented methodology, real-world datasets from the financial market and single-particle tracking experiments in biological gels are analyzed.