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Fractional Brownian motion, a Gaussian non-Markovian self-similar process with stationary long-correlated increments, has been identified to give rise to the anomalous diffusion behavior in a great variety of physical systems. The correlation and diffusion properties of this random motion are fully characterized by its index of self-similarity or the Hurst exponent. However, recent single-particle tracking experiments in biological cells revealed highly complicated anomalous diffusion phenomena that cannot be attributed to a class of self-similar random processes. Inspired by these observations, we here study the process that preserves the properties of the fractional Brownian motion at a single trajectory level; however, the Hurst index randomly changes from trajectory to trajectory. We provide a general mathematical framework for analytical, numerical, and statistical analysis of the fractional Brownian motion with the random Hurst exponent. The explicit formulas for probability density function, mean-squared displacement, and autocovariance function of the increments are presented for three generic distributions of the Hurst exponent, namely, two-point, uniform, and beta distributions. The important features of the process studied here are accelerating diffusion and persistence transition, which we demonstrate analytically and numerically. [ABSTRACT FROM AUTHOR]

We point out a minor mistake in Fig. 10 in the published version of our paper [M. Balcerek et al., Chaos 32, 093114 (2022)]. The conclusions drawn from the illustration remain the same. [ABSTRACT FROM AUTHOR]

In this paper, we introduce two tempered linear and non-linear time series models, namely, an autoregressive tempered fractionally integrated moving average (ARTFIMA) with α -stable noise and ARTFIMA with generalized autoregressive conditional heteroskedasticity (GARCH) noise (ARTFIMA-GARCH). We provide estimation procedures for the processes and explain the connection between ARTFIMA and their tempered continuous-time counterparts. Next, we demonstrate an application of the processes to modeling of heavy-tailed data from solar flare soft x-ray emissions. To this end, we study the solar flare data during a period of solar minimum, which occurred most recently in July, August, and September 2017. We use a two-state hidden Markov model to classify the data into two states (lower and higher activity) and to extract stationary trajectories. We do an end-to-end analysis and modeling of the solar flare data using both ARTFIMA and ARTFIMA-GARCH models and their non-tempered counterparts. We show through visual inspection and statistical tests that the ARTFIMA and ARTFIMA-GARCH models describe the data better than the ARFIMA and ARFIMA-GARCH, especially in the second state, which justifies that tempered processes can serve as the state-of-the-art approach to model signals originating from a power-law source with long memory effects. [ABSTRACT FROM AUTHOR]

Gaussian processes are powerful tools for modeling and predicting various numerical data. Hence, checking their quality of fit becomes a vital issue. In this article, we introduce a testing methodology for general Gaussian processes based on a quadratic form statistic. We illustrate the methodology on three statistical tests recently introduced in the literature, which are based on the sample autocovariance function, time average mean-squared displacement, and detrended moving average statistics. We compare the usefulness of the tests by taking into consideration three very important Gaussian processes: the fractional Brownian motion, which is self-similar with stationary increments (SSSIs), scaled Brownian motion, which is self-similar with independent increments (SSIIs), and the Ornstein–Uhlenbeck (OU) process, which is stationary. We show that the considered statistics' ability to distinguish between these Gaussian processes is high, and we identify the best performing tests for different scenarios. We also find that there is no omnibus quadratic form test; however, the detrended moving average test seems to be the first choice in distinguishing between same processes with different parameters. We also show that the detrended moving average method outperforms the Cholesky method. Based on the previous findings, we introduce a novel procedure of discriminating between Gaussian SSSI, SSII, and stationary processes. Finally, we illustrate the proposed procedure by applying it to real-world data, namely, the daily EURUSD currency exchange rates, and show that the data can be modeled by the OU process. [ABSTRACT FROM AUTHOR]