Stable Random Processes Research Articles

The intervals between zero-crossings of non-Gaussian stable random processes

Properties of the intervals between zero-crossings of non-Gaussian stable random processes are investigated. The classes of process considered are divided into those having short- and long-term memories, characterized by a coherence function that imbues dependence between a random variable measured at different times, which replaces the auto-correlation function, which exists only for the Gaussian case. For exponential coherence functions, all moments of probability densities for the intervals exist and a persistence parameter is determined that characterizes the rate of decay of the exponential tail of the probability densities, together with the full form of the density functions for selected values of the stability index. Results are verified with numerical simulation of a Cauchy process using a method independent of the theoretical development. For power-law coherence functions, the existence of moments of the distribution is conditional upon the indices characterizing the stable process and coherence function. The probability densities of the intervals have power-law tails depending on the product of these indices, with a logarithmic correction for specific combinations of these. An outer-scale or cut-off to power-law coherence functions regains a persistence parameter to the densities and the existence of all moments of the intervals.

Inference for the VEC(1) model with a heavy-tailed linear process errors*

This article studies the first-order vector error correction (VEC(1)) model when its noise is a linear process of independent and identically distributed (i.i.d.) heavy-tailed random vectors with a tail index . We show that the rate of convergence of the least squares estimator (LSE) related to the long-run parameters is n (sample size) and its limiting distribution is a stochastic integral in terms of two stable random processes, while the LSE related to the short-term parameters is not consistent. We further propose an automated approach via adaptive shrinkage techniques to determine the cointegrating rank in the VEC(1) model. It is demonstrated that the cointegration rank r 0 can be consistently selected despite the fact that the LSE related to the short-term parameters is not consistently estimable when the tail index . Simulation studies are carried out to evaluate the performance of the proposed procedure in finite samples. Last, we use our techniques to explore the long-run and short-run behavior of the monthly prices of wheat, corn, and wheat flour in the United States.

Novel VLSI Architecture for Fractional-Order Correntropy Adaptive Filtering Algorithm

Conventional adaptive filters, which assume Gaussian distribution for signal and noise, exhibit significant performance degradation when operating in non-Gaussian environments. Recently proposed fractional-order adaptive filters (FoAFs) address this concern by assuming that the signal and noise are symmetric <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\alpha $ </tex-math></inline-formula> -stable random processes. However, the literature does not include any VLSI architectures for these algorithms. Toward that end, this article develops hardware-efficient architecture for fractional-order correntropy adaptive filter (FoCAF). We first reformulate the FoCAF for its efficient real-time VLSI implementation and then demonstrate that these reformulations cause negligible performance degradation under the 16-bit fixed-point implementation. Using this reformulated algorithm, we design an FoCAF architecture. Furthermore, we analyze the critical path of the design to select the appropriate level of pipelining based on the sampling rate of the application. According to the critical-path analysis, the FoCAF design is pipelined using retiming techniques to obtain delayed FoCAF (DFoCAF), which is then synthesized using <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\mathbf {45}$ </tex-math></inline-formula> -nm CMOS technology. Synthesis results reveal that DFoCAF architecture requires a minimal increase in hardware over the prominent least mean square (LMS) filter architecture and achieves a significant increase in the performance in symmetric <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\alpha $ </tex-math></inline-formula> -stable environments where LMS fails to converge.

Dynamic Response Simulation of the Slope Triggered by Fluctuating Pressure during High Dam Flood Discharge

The problem of slope vibration induced by flood discharge of high dams cannot be ignored. In this paper, this problem is simplified to be the relationship among the fluctuating pressures, the inherent properties of the slope system, and its dynamic responses. The characteristics of fluctuating pressures in the plunge pool during flood discharge are analyzed by the hydraulic model test. And a numerical slope model consisting of the finite element domain and the PML domain is established to study the inherent characteristics and vibration responses by numerical simulation. The results show that the flow fluctuation in plunge pool is a continuous and stable random vibration process. And the fluctuating pressures on the bottom area of the plunge pool are greater than that of the sidewall, but the fluctuation phenomenon at the sidewall is more intense. In addition, the slope displacement vibration intensity caused by fluctuating pressures is small, only at the micron level. The fluctuating pressures on the bottom area of the plunge pool are the dominant factor affecting the kinetic energy density distribution of the slope. This work provides a simple and efficient method for the study of slope vibration induced by high dam flood discharge.

THE MAXIMUM PRINCIPLE FOR THE EQUATION OF LOCAL FLUCTUATIONS OF RIESZ GRAVITATIONAL FIELDS OF PURELY FRACTIONAL ORDER

The parabolic pseudodifferential equation with the Riesz fractional differentiation operator of α ∈ (0; 1) order, which acts on a spatial variable, is considered in the paper. This equation naturally summarizes the known equation of fractal diffusion of purely fractional order. It arises in the mathematical modeling of local vortices of nonstationary Riesz gravitational fields caused by moving objects, the interaction between the masses of which is characterized by the corresponding Riesz potential. The fundamental solution of the Cauchy problem for this equati- on is the density distribution of the probabilities of the force of local interaction between these objects, it belongs to the class of Polya distributions of symmetric stable random processes. Under certain conditions, for the coefficient of local field fluctuations, an analogue of the maximum principle was established for this equation. This principle is important in particular for substantiating the unity of the solution of the Cauchy problem on a time interval where the fluctuation coefficient is a non-decreasing function.

Long range dependence for stable random processes

We investigate long and short memory in ‐stable moving averages and max‐stable processes with ‐Fréchet marginal distributions. As these processes are heavy‐tailed, we rely on the notion of long range dependence based on the covariance of indicators of excursion sets. Sufficient conditions for the long and short range dependence of ‐stable moving averages are proven in terms of integrability of the corresponding kernel functions. For max‐stable processes, the extremal coefficient function is used to state a necessary and sufficient condition for long range dependence.

Fractional-Order Correntropy Filters for Tracking Dynamic Systems in α-Stable Environments

In an increasing number of modern filtering applications, the encountered signals consist of frequent sharp spikes, that cannot be accurately modeled using Gaussian random processes. Modeling the behavior of such signals requires the more general framework of $\alpha $ -stable random processes. In order to present an inclusive filtering solution, this brief derives a new class of fractional-order correntropy adaptive filters that are robust to the jittery $\alpha $ -stable signals. In contrast to conventional correntropy filters, the proposed objective function is compatible with the characteristic function of $\alpha $ -stable processes and captures fractional moments; therefore, the resulting algorithms do not depend on non-existing second-order moments. The work also includes performance and convergence analysis of the derived algorithms. Finally, simulations are conducted to illustrate the effectiveness of the proposed filtering techniques, which indicate that the proposed filters can outperform their counterparts and show less sensitivity to changes in the $\alpha $ parameter.

ON THE NATURE OF A CLASSICAL PSEUDODIFFERENTIAL EQUATION

The work is devoted to the study of the general nature of one classical parabolic pseudodi- fferential equation with the operator M.Rice of fractional differentiation. At the corresponding values of the order of fractional differentiation, this equation is also known as the isotropic superdiffusion equation. It is a natural generalization of the classical diffusion equation. It is also known that the fundamental solution of the Cauchy problem for this equation is the density distribution of probabilities of stable symmetric random processes by P.Levy. The paper shows that the fundamental solution of this equation is the distribution of probabilities of the force of local influence of moving objects in a nonstationary gravitational field, in which the interaction between masses is subject to the corresponding potential of M.Rice. In this case, the classical case of Newton’s gravity corresponds to the known nonstationary J.Holtsmark distribution.

Inference in heavy-tailed vector error correction models

This paper first studies the full rank least squares estimator (FLSE) of the heavy-tailed vector error correction (VEC) models. It is shown that the rate of convergence of the FLSE related to the long-run parameters is n (sample size) and its limiting distribution is a stochastic integral in terms of two stable random processes when the tail index α∈(0,2). Furthermore, we show that the rate of convergence of the FLSE related to the short-term parameters is n1∕αL̃(n) and its limiting distribution is a functional of two stable processes when α∈(1,2), where L̃(n) is a slowly varying function. However, when α∈(0,1), we show that the rate of convergence of the FLSE related to the short-term parameters is n and its limiting distribution not only depends on the stationary component itself but also depends on the unit root component. Based on the FLSE, we then study the limiting behavior of the reduced rank LSE (RLSE). The results related to the short-term parameters of both FLSE and RLSE are significantly different from those of heavy-tailed time series in the literature, and it may provide new insights in the area for future research. Simulation study is carried out to demonstrate the performance of both estimators. A real example with application to 3-month Treasury Bill rate, 1-year Treasury Bill rate and Federal Fund rate is given.

On the crossings number of a hyperplane by a stable random process

The numbers of crossings of a hyperplane by discrete approximations for trajectories of an $\alpha$-stable random process (with $1&lt;\alpha&lt;2$) and some processes related to it are investigated. We consider an $\alpha$-stable process is killed with some intensity on the hyperplane and a pseudo-process that is formed from the $\alpha$-stable process using its perturbation by a fractional derivative operator with a multiplier like a delta-function on the hyperplane. In each of these cases, the limit distribution of the crossing number of the hyperplane by some discret approximation of the process is related to the distribution of its local time on this hyperplane. Integral equations for characteristic functions of these distributions are constructed. Unique bounded solutions of these equations can be constructed by the method of successive approximations.

One Dimensional Random Motion on Segment with Reflecting Edges and Dependent Increments

In previous papers, the author considered the model of anomalous diffusion, defined by stable random process on an interval with reflecting edges. Estimates of the rate convergence of this process distribution to a uniform distribution are constructed. However, recent physical studies require consideration of models of diffusion, defined not only by stable random process with independent increments but multivariate fractional Brownian motion with dependent increments. This task requires the development of special mathematical techniques evaluation of the rate of convergence of the distribution of multivariate Brownian motion in a segment with reflecting boundaries to the limit. In the present work, this technology is developed and a power estimate of the rate of convergence to the limiting uniform distribution is built.

First-passage problem for nonlinear systems under Lévy white noise through path integral method

In this paper, the first-passage problem for nonlinear systems driven by \(\alpha \)-stable Levy white noises is considered. The path integral solution (PIS) is adopted for determining the reliability function and first-passage time probability density function of nonlinear oscillators. Specifically, based on the properties of \(\alpha \)-stable random variables and processes, PIS is extended to deal with Levy white noises with any value of the stability index \(\alpha \). Application to linear and nonlinear systems considering different values of \(\alpha \) is reported. Comparisons with pertinent Monte Carlo simulation data demonstrate the accuracy of the results.

Stable random fields, point processes and large deviations

We investigate the large deviation behaviour of a point process sequence based on a stationary symmetric α-stable (0<α<2) discrete-parameter random field using the framework of Hult and Samorodnitsky (2010). Depending on the ergodic theoretic and group theoretic structures of the underlying nonsingular group action, we observe different large deviation behaviours of this point process sequence. We use our results to study the large deviations of various functionals (e.g., partial sum, maxima, etc.) of stationary symmetric stable fields.

On the joint statistics of stable random processes

A utilitarian continuous bi-variate random process whose first-order probability density function is a stable random variable is constructed. Results paralleling some of those familiar from the theory of Gaussian noise are derived. In addition to the joint-probability density for the process, these include fractional moments and structure functions. Although the correlation functions for stable processes other than Gaussian do not exist, we show that there is coherence between values adopted by the process at different times, which identifies a characteristic evolution with time. The distribution of the derivative of the process, and the joint-density function of the value of the process and its derivative measured at the same time are evaluated. These enable properties to be calculated analytically such as level crossing statistics and those related to the random telegraph wave. When the stable process is fractal, the proportion of time it spends at zero is finite and some properties of this quantity are evaluated, an optical interpretation for which is provided.

Small deviations of stable processes and entropy of the associated random operators

We investigate the relation between the small deviation problem for a symmetric $\alpha$-stable random vector in a Banach space and the metric entropy properties of the operator generating it. This generalizes former results due to Li and Linde and to Aurzada. It is shown that this problem is related to the study of the entropy numbers of a certain random operator. In some cases, an interesting gap appears between the entropy of the original operator and that of the random operator generated by it. This phenomenon is studied thoroughly for diagonal operators. Basic ingredients here are techniques related to random partitions of the integers. The main result concerning metric entropy and small deviations allows us to determine or provide new estimates for the small deviation rate for several symmetric $\alpha$-stable random processes, including unbounded Riemann--Liouville processes, weighted Riemann--Liouville processes and the ($d$-dimensional) $\alpha$-stable sheet.

Best linear prediction for [formula omitted]-stable random processes

The best linear prediction for α -stable random processes based on some past values is presented. The prediction is the best with respect to a criterion known as stable covariation. The minimum stable covariations can be considered as the smallest error tail probabilities. The predictor obtained is equal to the best linear predictor based on minimization of second-moment error for Gaussian processes.

Stochastic analysis of external and parametric dynamical systems under sub-Gaussian Levy white-noise

In this study stochastic analysis of non-linear dynamical systems under ?-stable, multiplicative white noise has been conducted. The analysis has dealt with a special class of ?-stable stochastic processes namely sub-Gaussian white noises. In this setting the governing equation either of the probability density function or of the characteristic function of the dynamical response may be obtained considering the dynamical system forced by a Gaussian white noise with an uncertain factor with ?/2-stable distribution. This consideration yields the probability density function or the characteristic function of the response by means of a simple integral involving the probability density function of the system under Gaussian white noise and the probability density function of the ?/2-stable random parameter. Some numerical applications have been reported assessing the reliability of the proposed formulation. Moreover a proper way to perform digital simulation of the sub-Gaussian ?-stable random process preventing dynamical systems from numerical overflows has been reported and discussed in detail.

Data block adaptive filtering algorithms for α-stable random processes

The least mean p-norm (LMP) algorithm is an effective algorithm for processing the signal of @a-stable distribution. This paper proposes data block adaptive filtering algorithms for the @a-stable random processes based on the fractional lower order statistics (FLOS). The data block algorithms change the direction of coefficient increment vector by introducing a matrix which includes the information of more past input signal vectors than which are used in the LMP algorithm during the iteration process, taking full advantage of the past values of the gradient vector during the adaptation. Simulations studies indicate that the proposed algorithms increase convergence rate in non-Gaussian stable distribution noise environments compared to the existing algorithms based on FLOS summarized in this paper.

Fast algorithms for simulation of symmetric stable random quantities and processes

A limit representation has been obtained for a uniform symmetric stable process with independent increments in the form of a sum of independent Wiener’s and purely jump-like generalized Gaussian processes. Simple and speedy algorithms are developed for simulation of symmetric stable random quantities and processes. The algorithms can be easily implemented in a computer.

Generation and monitoring of discrete stable random processes using multiple immigration population models

Some properties of classical population processes that comprise births, deaths and multiple immigrations are investigated. The rates at which the immigrants arrive can be tailored to produce a population whose steady state fluctuations are described by a pre-selected distribution. Attention is focused on the class of distributions with a discrete stable law, which have power-law tails and whose moments and autocorrelation function do not exist. The separate problem of monitoring and characterizing the fluctuations is studied, analysing the statistics of individuals that leave the population. The fluctuations in the size of the population are transferred to the times between emigrants that form an intermittent time series of events. The emigrants are counted with a detector of finite dynamic range and response time. This is modelled through clipping the time series or saturating it at an arbitrary but finite level, whereupon its moments and correlation properties become finite. Distributions for the time to the first counted event and for the time between events exhibit power-law regimes that are characteristic of the fluctuations in population size. The processes provide analytical models with which properties of complex discrete random phenomena can be explored, and in addition provide generic means by which random time series encompassing a wide range of intermittent and other discrete random behaviour may be generated.