Fractional Gaussian Processes Research Articles

Quantile estimation in fractional Levy Ornstein-Uhlenbeck processes

First we study estimation of the drift parameter in the fractional Ornstein-Uhlenbeck process whose marginal distribution is Student t-distribution. We obtain Spearman’s correlation based estimator, quantile estimator and Brownian excursion based estimator of the drift parameter. Then we study method of moments estimator and quantile estimator in fractional inverse Gaussian and fractional gamma Ornstein-Uhlenbeck processes.

Properties of Various Entropies of Gaussian Distribution and Comparison of Entropies of Fractional Processes

We consider five types of entropies for Gaussian distribution: Shannon, Rényi, generalized Rényi, Tsallis and Sharma–Mittal entropy, establishing their interrelations and their properties as the functions of parameters. Then, we consider fractional Gaussian processes, namely fractional, subfractional, bifractional, multifractional and tempered fractional Brownian motions, and compare the entropies of one-dimensional distributions of these processes.

On the Genuine Relevance of the Data-Driven Signal Decomposition-Based Multiscale Permutation Entropy.

Ordinal pattern-based approaches have great potential to capture intrinsic structures of dynamical systems, and therefore, they continue to be developed in various research fields. Among these, the permutation entropy (PE), defined as the Shannon entropy of ordinal probabilities, is an attractive time series complexity measure. Several multiscale variants (MPE) have been proposed in order to bring out hidden structures at different time scales. Multiscaling is achieved by combining linear or nonlinear preprocessing with PE calculation. However, the impact of such a preprocessing on the PE values is not fully characterized. In a previous study, we have theoretically decoupled the contribution of specific signal models to the PE values from that induced by the inner correlations of linear preprocessing filters. A variety of linear filters such as the autoregressive moving average (ARMA), Butterworth, and Chebyshev were tested. The current work is an extension to nonlinear preprocessing and especially to data-driven signal decomposition-based MPE. The empirical mode decomposition, variational mode decomposition, singular spectrum analysis-based decomposition and empirical wavelet transform are considered. We identify possible pitfalls in the interpretation of PE values induced by these nonlinear preprocessing, and hence, we contribute to improving the PE interpretation. The simulated dataset of representative processes such as white Gaussian noise, fractional Gaussian processes, ARMA models and synthetic sEMG signals as well as real-life sEMG signals are tested.

Spectral asymptotics for a class of integro-differential equations arising in the theory of fractional Gaussian processes

We study spectral problems for integro-differential equations arising in the theory of Gaussian processes similar to the fractional Brownian motion. We generalize the method of Chigansky–Kleptsyna and obtain the two-term eigenvalue asymptotics for such equations. Application to the small ball probabilities in [Formula: see text]-norm is given.

Continuous Breuer–Major theorem: Tightness and nonstationarity

Let $Y=(Y(t))_{t\geq 0}$ be a zero-mean Gaussian stationary process with covariance function $\rho :\mathbb{R}\to \mathbb{R}$ satisfying $\rho (0)=1$. Let $f:\mathbb{R}\to \mathbb{R}$ be a square-integrable function with respect to the standard Gaussian measure, and suppose the Hermite rank of $f$ is $d\geq 1$. If $\int_{\mathbb{R}}|\rho (s)|^{d}\,ds<\infty $, then the celebrated Breuer–Major theorem (in its continuous version) asserts that the finite-dimensional distributions of $Z_{\varepsilon }:=\sqrt{\varepsilon }\int_{0}^{\cdot /\varepsilon }f(Y(s))\,ds$ converge to those of $\sigma W$ as $\varepsilon \to 0$, where $W$ is a standard Brownian motion and $\sigma $ is some explicit constant. Since its first appearance in 1983, this theorem has become a crucial probabilistic tool in different areas, for instance in signal processing or in statistical inference for fractional Gaussian processes. The goal of this paper is twofold. First, we investigate the tightness in the Breuer–Major theorem. Surprisingly, this problem did not receive a lot of attention until now, and the best available condition due to Ben Hariz [J. Multivariate Anal. 80 (2002) 191–216] is neither arguably very natural, nor easy-to-check in practice. In contrast, our condition very simple, as it only requires that $|f|^{p}$ must be integrable with respect to the standard Gaussian measure for some $p$ strictly bigger than 2. It is obtained by means of the Malliavin calculus, in particular Meyer inequalities. Second, and motivated by a problem of geometrical nature, we extend the continuous Breuer–Major theorem to the notoriously difficult case of self-similar Gaussian processes which are not necessarily stationary. An application to the fluctuations associated with the length process of a regularized version of the bifractional Brownian motion concludes the paper.

Fluid heterogeneity detection based on the asymptotic distribution of the time-averaged mean squared displacement in single particle tracking experiments

A tracer particle is called anomalously diffusive if its mean squared displacement grows approximately as as a function of time t for some constant , where the diffusion exponent satisfies . In this article, we use recent results on the asymptotic distribution of the time-averaged mean squared displacement (Didier and Zhang 2017 J. Time Ser. Anal. 38 395–416) to construct statistical tests for detecting physical heterogeneity in viscoelastic fluid samples starting from one or multiple observed anomalously diffusive paths. The methods are asymptotically valid for the range and involve a mathematical characterization of time-averaged mean squared displacement bias and the effect of correlated disturbance errors. The assumptions on particle motion cover a broad family of fractional Gaussian processes, including fractional Brownian motion and many fractional instances of the generalized Langevin equation framework. We apply the proposed methods in experimental data from treated Pseudomonas aeruginosa biofilms generated by the collaboration of the Hill and Schoenfisch Labs at UNC-Chapel Hill.

Least squares estimation for the drift parameters in the sub-fractional Vasicek processes

While the statistical inference of Vasicek processes driven by both Brownian motions and fractional Brownian motions has a long history, the statistical analysis for the Vasicek model driven by other fractional Gaussian processes is obviously more recent. This paper considers the parameter estimation problem for Vasicek processes driven by sub-fractional Brownian motions with the known Hurst parameter greater than one half. Since the maximum likelihood estimators are hardly analyzed because of the stochastic integrals with singular kernels, least squares estimators for drift parameters are provided based on time-continuous observations. The strong consistency results as well as the asymptotic distributions of these estimators are obtained in both the non-ergodic case and the null recurrent case.

Arbitrage with fractional Gaussian processes

While the arbitrage opportunity in the Black–Scholes model driven by fractional Brownian motion has a long history, the arbitrage strategy in the Black–Scholes model driven by general fractional Gaussian processes is in its infancy. The development of stochastic calculus with respect to fractional Gaussian processes allowed us to study such models. In this paper, following the idea of Shiryaev (1998), an arbitrage strategy is constructed for the Black–Scholes model driven by fractional Gaussian processes, when the stochastic integral is interpreted in the Riemann–Stieltjes sense. Arbitrage opportunities in some fractional Gaussian processes, including fractional Brownian motion, sub-fractional Brownian motion, bi-fractional Brownian motion, weighted-fractional Brownian motion and tempered fractional Brownian motion, are also investigated.

Some fractional and multifractional Gaussian processes: A brief introduction

This paper gives a brief introduction to some important fractional and multifractional Gaussian processes commonly used in modelling natural phenomena and man-made systems. The processes include fractional Brownian motion (both standard and the Riemann-Liouville type), multifractional Brownian motion, fractional and multifractional Ornstein-Uhlenbeck processes, fractional and mutifractional Reisz-Bessel motion. Possible applications of these processes are briefly mentioned.

Quantum probes for fractional Gaussian processes

We address the characterization of classical fractional random noise via quantum probes. In particular, we focus on estimation and discrimination problems involving the fractal dimension of the trajectories of a system subject to fractional Brownian noise. We assume that the classical degree of freedom exposed to the environmental noise is coupled to a quantum degree of freedom of the same system, e.g. its spin, and exploit quantum limited measurements on the spin part to characterize the classical fractional noise. More generally, our approach may be applied to any two-level system subject to dephasing perturbations described by fractional Brownian noise, in order to assess the precision of quantum limited measurements in the characterization of the external noise. In order to assess the performances of quantum probes we evaluate the Bures metric, as well as the Helstrom and the Chernoff bound, and optimize their values over the interaction time. We find that quantum probes may be successfully employed to obtain a reliable characterization of fractional Gaussian process when the coupling with the environment is weak or strong. In the first case decoherence is not much detrimental and for long interaction times the probe acquires information about the environmental parameters without being too much mixed. Conversely, for strong coupling information is quickly impinged on the quantum probe and can be effectively retrieved by measurements performed in the early stage of the evolution. In the intermediate situation, none of the two above effects take place: information is flowing from the environment to the probe too slowly compared to decoherence, and no measurements can be effectively employed to extract it from the quantum probe. The two regimes of weak and strong coupling are defined in terms of a threshold value of the coupling, which itself increases with the fractional dimension.

Series Representations of Fractional Gaussian Processes by Trigonometric and Haar Systems

The aim of the present paper is to investigate series representations of the Riemann-Liouville process $R^\alpha$, $\alpha >1/2$, generated by classical orthonormal bases in $L_2[0,1]$. Those bases are, for example, the trigonometric or the Haar system. We prove that the representation of $R^\alpha$ via the trigonometric system possesses the optimal convergence rate if and only if $1/2 3/2$ a representation via the Haar system is not optimal. Estimates for the rate of convergence of the Haar series are given in the cases $\alpha > 3/2$ and $\alpha = 3/2$. However, in this latter case the question whether or not the series representation is optimal remains open. Recently M. A. Lifshits answered this question (cf. [13]). Using a different approach he could show that in the case $\alpha = 3/2$ a representation of the Riemann-Liouville process via the Haar system is also not optimal.

On the Lower Classes of Some Mixed Fractional Gaussian Processes with Two Logarithmic Factors

We introduce the fractional mixed fractional Brownian sheet and investigate the small ball behavior of its sup‐norm statistic by establishing a general result on the small ball probability of the sum of two not necessarily independent joint Gaussian random vectors. Then, we state general conditions and characterize the sufficiency part of the lower classes of some statistics of the above process by an integral test. Finally, when we consider the sup‐norm statistic, the necessity part is given by a second integral test.

Singularity functions for fractional processes: application to the fractional Brownian sheet

In this paper almost sure convergence and asymptotic normality of generalized quadratic variation are studied. The main result in this paper extend classical results from Baxter and Gladyshev so that they can be applied to fractional Gaussian processes. An application to the estimation of the true axes of a fractional Brownian sheet is also obtained.

Logarithmic L2-Small Ball Asymptotics for some Fractional Gaussian Processes

We find the logarithmic L2-small ball asymptotics of some Gaussian processes related to the fractional Brownian motion (fBm), fractional Ornstein--Uhlenbeck process (fOU), and their integrated analogues. We consider also the multiparameter generalizations.

Penalised maximum likelihood estimation for fractional Gaussian processes

We apply and extend Firth's (1993) modified score estimator to deal with a class of stationary Gaussian long-memory processes. Our estimator removes the first-order bias of the maximum likelihood estimator. A small simulation study reveals the reduction in the bias is considerable, while it does not inflate the corresponding mean squared error.