Linear Fractional Stable Motion Research Articles

Fractionally Integrated Moving Average Stable Processes With Long-Range Dependence

Long memory processes driven by L\'evy noise with finite second-order moments have been well studied in the literature. They form a very rich class of processes presenting an autocovariance function which decays like a power function. Here, we study a class of L\'evy process whose second-order moments are infinite, the so-called $\alpha$-stable processes. Based on Samorodnitsky and Taqqu (2000), we construct an isometry that allows us to define stochastic integrals concerning the linear fractional stable motion using Riemann-Liouville fractional integrals. With this construction, follows naturally an integration by parts formula. We then present a family of stationary $S\alpha S$ processes with the property of long-range dependence, using a generalized measure to investigate its dependence structure. In the end, the law of large number's result for a time's sample of the process is shown as an application of the isometry and integration by parts formula.

Multidimensional parameter estimation of heavy‐tailed moving averages

AbstractIn this article we present a parametric estimation method for certain multiparameter heavy‐tailed Lévy‐driven moving averages. The theory relies on recent multivariate central limit theorems obtained via Malliavin calculus on Poisson spaces. Our minimal contrast approach is related to previous papers, which propose to use the marginal empirical characteristic function to estimate the one‐dimensional parameter of the kernel function and the stability index of the driving Lévy motion. We extend their work to allow for a multiparametric framework that in particular includes the important examples of the linear fractional stable motion, the stable Ornstein–Uhlenbeck process, certain CARMA(2, 1) models, and Ornstein–Uhlenbeck processes with a periodic component among other models. We present both the consistency and the associated central limit theorem of the minimal contrast estimator. Furthermore, we demonstrate numerical analysis to uncover the finite sample performance of our method.

A minimal contrast estimator for the linear fractional stable motion

In this paper we present an estimator for the three-dimensional parameter $$(\sigma , \alpha , H)$$ of the linear fractional stable motion, where H represents the self-similarity parameter, and $$(\sigma , \alpha )$$ are the scaling and stability parameters of the driving symmetric Levy process L. Our approach is based upon a minimal contrast method associated with the empirical characteristic function combined with a ratio type estimator for the self-similarity parameter H. The main result investigates the strong consistency and weak limit theorems for the resulting estimator. Furthermore, we propose several ideas to obtain feasible confidence regions in various parameter settings. Our work is mainly related to Ljungdahl and Podolskij (A note on parametric estimation of Levy moving average processes, p 294, 2019) and Mazur et al. (Bernoulli 26(1): 226–252, 2020) in which parameter estimation for the linear fractional stable motion and related Levy moving average processes has been studied.

Linear Fractional Stable Motion with the rlfsm R Package

Linear fractional stable motion is a type of a stochastic integral driven by symmetric alpha-stable L´evy motion. The integral could be considered as a non-Gaussian analogue of the fractional Brownian motion. The present paper discusses R package rlfsm created for numerical procedures with the linear fractional stable motion. It is a set of tools for simulation of these processes as well as performing statistical inference and simulation studies on them. We introduce: tools that we developed to work with that type of motions as well as methods and ideas underlying them. Also we perform numerical experiments to show finite-sample behavior of certain estimators of the integral, and give an idea of how to envelope workflow related to the linear fractional stable motion in S4 classes and methods. Supplementary materials, including codes for numerical experiments, are available online. rlfsm could be found on CRAN and gitlab.

A note on linear processes with tapered innovations

In the paper, we consider the partial-sum process \( {\sum}_{k=1}^{\left[ nt\right]}{X}_k^{(n)}, \) where \( \left\{{X}_n^{(n)},k\in \mathbb{Z}\right\},n\ge 1, \) is a series of linear processes with innovations having heavy-tailed tapered distributions with tapering parameter bn depending on n. We show that, depending on the properties of a filter of a linear process under consideration and on the parameter bn defining if the tapering is hard or soft, the limit process for such partial-sum process can be a fractional Brownian motion or linear fractional stable motion.

Estimation of the linear fractional stable motion

In this paper, we investigate the parametric inference for the linear fractional stable motion in high and low frequency setting. The symmetric linear fractional stable motion is a three-parameter family, which constitutes a natural non-Gaussian analogue of the scaled fractional Brownian motion. It is fully characterised by the scaling parameter $\sigma>0$, the self-similarity parameter $H\in(0,1)$ and the stability index $\alpha\in(0,2)$ of the driving stable motion. The parametric estimation of the model is inspired by the limit theory for stationary increments Lévy moving average processes that has been recently studied in (Ann. Probab. 45 (2017) 4477–4528). More specifically, we combine (negative) power variation statistics and empirical characteristic functions to obtain consistent estimates of $(\sigma,\alpha,H)$. We present the law of large numbers and some fully feasible weak limit theorems.

Estimation of the Hurst and the stability indices of a $H$-self-similar stable process

In this paper we estimate both the Hurst and the stability indices of a $H$-self-similar stable process. More precisely, let $X$ be a $H$-sssi (self-similar stationary increments) symmetric $\alpha$-stable process. The process $X$ is observed at points $\frac{k}{n}$, $k=0,\ldots,n$. Our estimate is based on $\beta$-negative power variations with $-\frac{1}{2}<\beta<0$. We obtain consistent estimators, with rate of convergence, for several classical $H$-sssi $\alpha$-stable processes (fractional Brownian motion, well-balanced linear fractional stable motion, Takenaka’s process, Levy motion). Moreover, we obtain asymptotic normality of our estimators for fractional Brownian motion and Levy motion.

Higher Order Fractional Stable Motion: Hyperdiffusion with Heavy Tails

We introduce the class of higher order fractional stable motions that can exhibit hyperdiffusive spreading with heavy tails. We define the class as a generalization of higher order fractional Brownian motion as well as a generalization of linear fractional stable motions. Higher order fractional stable motions are self-similar with Hurst index larger than one and non-Gaussian stable marginals with infinite variance and have stationary higher order increments. We investigate their sample path properties and asymptotic dependence structure on the basis of codifference. In particular, by incrementing or decrementing sample paths once under suitable conditions, the diffusion rate can be accelerated or decelerated by one order. With a view towards simulation study, we provide a ready-for-use sample path simulation recipe at discrete times along with error analysis. The proposed simulation scheme requires only elementary numerical operations and is robust to high frequency sampling, irregular spacing and super-sampling.

A novel crowd flow model based on linear fractional stable motion

For the evacuation dynamics in indoor space, a novel crowd flow model is put forward based on Linear Fractional Stable Motion. Based on position attraction and queuing time, the calculation formula of movement probability is defined and the queuing time is depicted according to linear fractal stable movement. At last, an experiment and simulation platform can be used for performance analysis, studying deeply the relation among system evacuation time, crowd density and exit flow rate. It is concluded that the evacuation time and the exit flow rate have positive correlations with the crowd density, and when the exit width reaches to the threshold value, it will not effectively decrease the evacuation time by further increasing the exit width.

A uniform law for convergence to the local times of linear fractional stable motions

We provide a uniform law for the weak convergence of additive functionals of partial sum processes to the local times of linear fractional stable motions, in a setting sufficiently general for statistical applications. Our results are fundamental to the analysis of the global properties of nonparametric estimators of nonlinear statistical models that involve such processes as covariates.

Linear Multifractional Stable Motion: Wavelet Estimation of H(·) and α Parameters*

Stoev and Taqqu introduced a linear multifractional stable motion (LMSM), an extension of a linear fractional stable motion (LFSM) such that the Hurst parameter H becomes a function H(t). The stability parameter α determines tail heaviness of marginal distributions of LMSM. Under some conditions, Stoev and Taqqu showed that H(t 0) is its self-similarity exponent at a t 0 ≠ 0; also, recently, Ayache and Hamonier established that H(t 0)−1/α and min t∈I H(t)−1/α are its local Hölder exponent at t 0 and uniform Hölder exponent on a compact interval I. We construct, strongly consistent wavelet estimators of min t∈I H(t), H(t 0), and α when α ∈ (1, 2) and H(·) is smooth with values in $$ \left[\underset{\bar{\mkern6mu}}{H},\overline{H}\right]\subset \left(1/\alpha, 1\right) $$ .

Linear multifractional stable motion: fine path properties

For at least a decade, there has been considerable interest in applied and theoretical issues related to multifractional random models. Nonetheless, only a few results are known in the framework of heavy-tailed stable distributions. In this framework, a paradigmatic example is the linear multifractional stable motion (LMSM) ${Y(t)\colon t\in\mathbb R}$. Stoev and Taqqu \[30, 29| introduced LMSM by replacing the constant Hurst parameter of classical linear fractional stable motion (LFSM) by a deterministic function $H(\cdot)$ depending on the time variable $t$. The main goal of our article is to make a comprehensive study of the local and asymptotic behavior of ${Y(t):t\in\mathbb R}$. To this end, one needs to derive fine path properties of ${X(u,v)\colon (u,v)\in\mathbb R \times (1/\alpha,1)}$, the field generating the process (i.e., one has $Y(t)=X(t,H(t))$ for all $t\in\mathbb R$). This leads us to introduce random wavelet series representations of ${X(u,v)\colon (u,v)\in\mathbb R \times (1/\alpha,1)}$ as well as of all its pathwise partial derivatives of any order with respect to $v$. Then our strategy consists in using wavelet methods which are reminiscent of those in \[2, 5]. Among other things, we solve a conjecture of Stoev and Taqqu concerning the existence for LMSM of a version with almost surely continuous paths; moreover we significantly improve Theorem 4.1 in \[29], which provides some bounds for the local H\\"older exponent (in other words, the uniform pointwise Hölder exponent) of LMSM. Namely, we obtain a quasi-optimal global modulus of continuity, and also an optimal local one. It is worth noticing that, even in the quite classical case of LFSM, the optimal local modulus of continuity provides a new result which was previously not known.

Tempered Fractional Stable Motion

Tempered fractional stable motion adds an exponential tempering to the power-law kernel in a linear fractional stable motion, or a shift to the power-law filter in a harmonizable fractional stable motion. Increments from a stationary time series that can exhibit semi-long-range dependence. This paper develops the basic theory of tempered fractional stable processes, including dependence structure, sample path behavior, local times, and local nondeterminism.

Scaling Properties of the Empirical Structure Function of Linear Fractional Stable Motion and Estimation of Its Parameters

Linear fractional stable motion is an example of a self-similar stationary increments stochastic process exhibiting both long-range dependence and heavy-tails. In this paper we propose methods that are able to estimate simultaneously the self-similarity parameter and the tail parameter. These methods are based on the asymptotic behavior of the so-called “empirical structure function”, a statistic which resembles a sample moment of the process. We show and use the fact that the rate of growth of the empirical structure function is determined by the Hurst parameter and the tail index. We test the methods on simulated data and apply them to network traffic and solar flares data

Fine regularity of Lévy processes and linear (multi)fractional stable motion

In this work, we investigate the fine regularity of Levy processes using the 2-microlocal formalism. This framework allows us to refine the multifractal spectrum determined by Jaffard and, in addition, study the oscillating singularities of Levy processes. The fractal structure of the latter is proved to be more complex than the classic multifractal spectrum and is determined in the case of alpha-stable processes. As a consequence of these fine results and the properties of the 2-microlocal frontier, we are also able to completely characterise the multifractal nature of the linear fractional stable motion (extension of fractional Brownian motion to α-stable measures) in the case of continuous and unbounded sample paths as well. The regularity of its multifractional extension is also presented, indirectly providing an example of a stochastic process with a non-homogeneous and random multifractal spectrum.

Linear fractional stable motion: A wavelet estimator of the [formula omitted] parameter

Linear fractional stable motion, denoted by {XH,α(t)}t∈R, is one of the most classical stable processes; it depends on two parameters H∈(0,1) and α∈(0,2). The parameter H characterizes the self-similarity property of {XH,α(t)}t∈R while the parameter α governs the tail heaviness of its finite dimensional distributions; throughout our article we assume that the latter distributions are symmetric, that H>1/α and that H is known. We show that, on the interval [0,1], the asymptotic behavior of the maximum, at a given scale j, of absolute values of the wavelet coefficients of {XH,α(t)}t∈R, is of the same order as 2−j(H−1/α); then we derive from this result a strongly consistent (i.e. almost surely convergent) statistical estimator for the parameter α.

Robustness of estimators of long-range dependence and self-similarity under non-Gaussianity

Long-range dependence (LRD) and non-Gaussianity are ubiquitous in many natural systems such as ecosystems, biological systems and climate. However, it is not always appreciated that the two phenomena may occur together in natural systems and that self-similarity in a system can be a superposition of both phenomena. These features, which are common in complex systems, impact the attribution of trends and the occurrence and clustering of extremes. The risk assessment of systems with these properties will lead to different outcomes (e.g. return periods) than the more common assumption of independence of extremes. Two paradigmatic models are discussed that can simultaneously account for LRD and non-Gaussianity: autoregressive fractional integrated moving average (ARFIMA) and linear fractional stable motion (LFSM). Statistical properties of estimators for LRD and self-similarity are critically assessed. It is found that the most popular estimators can be biased in the presence of important features of many natural systems like trends and multiplicative noise. Also the LRD and non-Gaussianity of two typical natural time series are discussed.

On the Fourier coefficients of linear fractional stable motion

Inspired by a theorem of Marcinkiewicz [J. Marcinkiewicz, On a class of functions and their Fourier series, C. R. Soc. Sci. Varsovie, 26:71–77, 1934. Reprinted in: J. Marcinkiewicz, Collected Papers (A. Zygmund (Ed.)), PanstwoweWydawnictwo Naukowe,Warsaw, 1964] stating that the maximum of the absolute values of real Fourier coefficients an and bn of a function of bounded p-variation \( \left( {p \geqslant 1} \right) \) on an interval [0, 1] is of order O(n−1/p) as n → ∞, we compute the Fourier coefficients of the linear fractional stable motion (LFSM) and of the closely related Riemann–Liouville (RL) process and investigate the rate of their decay.

A strong law of large numbers with applications to self-similar stable processes

Let p ∈(0,∞) be a constant and let ξn ⊂ Lp(Σ,F,ℙ) be a sequence of random variables. For any integers m, n ≥ 0, denote \(S_{m,n}=\sum{_{k=m}^{m+n-1}}\xi_k\). It is proved that, if there exist a nondecreasing function ϕ ℝ+ → ℝ+ (which satisfies a mild regularity condition) and an appropriately chosen integer a ≥ 2 such that $$\sum_{n=0}^\infty\begin{array}{c}\sup\mathbb{E}\\ {k \geq 0}\end{array}|\frac{S_{{k,a}^n}}{\varphi (a^n)}|<\infty$$ then $$\lim_{n \rightarrow \infty}\frac{S_{0,n}}{\varphi(n)}=0$$ This extends Theorem 1 in Chobanyan, Levental and Salehi [3] and can be applied conveniently to a wide class of self-similar processes with stationary increments including stable processes.

A SIMULATION METHOD FOR LOG-FRACTIONAL STABLE MOTION

Stoev and Taqqu1 presented a fast simulation method for the linear fractional stable motion (lfsm) process, using the fast Fourier transform. In this paper, we extend this method for fast simulation of the log-fractional stable motion (log-fsm) process. These two heavy-tailed processes, i.e. lfsm and log-fsm, can be used for modeling the modern telecommunication networks. In addition, a necessary and sufficient condition is given for the efficiency of this method. Finally, an upper bound for approximation error is presented.