non-Gaussian Stable 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.

Convergence of partial sum processes to stable processes with application for aggregation of branching processes

We provide a generalization of Theorem 1 in Bartkiewicz et al. (2011) in the sense that we give sufficient conditions for weak convergence of finite dimensional distributions of the partial sum processes of a strongly stationary sequence to the corresponding finite dimensional distributions of a non-Gaussian stable process instead of weak convergence of the partial sums themselves to a non-Gaussian stable distribution. As an application, we describe the asymptotic behaviour of finite dimensional distributions of aggregation of independent copies of a strongly stationary subcritical Galton–Watson branching process with regularly varying immigration having index in (0,1)∪(1,4/3) in a so-called iterated case, namely when first taking the limit as the time scale and then the number of copies tend to infinity.

Efficient estimation of stable Lévy process with symmetric jumps

Efficient estimation of a non-Gaussian stable Levy process with drift and symmetric jumps observed at high frequency is considered. For this statistical experiment, the local asymptotic normality of the likelihood is proved with a non-singular Fisher information matrix through the use of a non-diagonal norming matrix. The asymptotic normality and efficiency of a sequence of roots of the associated likelihood equation are shown as well. Moreover, we show that a simple preliminary method of moments can be used as an initial estimator of a scoring procedure, thereby conveniently enabling us to bypass numerically demanding likelihood optimization. Our simulation results show that the one-step estimator can exhibit quite similar finite-sample performance as the maximum likelihood estimator.

Futures pricing in electricity markets based on stable CARMA spot models

We present a new model for the electricity spot price dynamics, which is able to capture seasonality, low-frequency dynamics and extreme spikes in the market. Instead of the usual purely deterministic trend we introduce a non-stationary independent increment process for the low-frequency dynamics, and model the large fluctuations by a non-Gaussian stable CARMA process. The model allows for analytic futures prices, and we apply these to model and estimate the whole market consistently. Besides standard parameter estimation, an estimation procedure is suggested, where we fit the non-stationary trend using futures data with long time until delivery, and a robust L1-filter to find the states of the CARMA process. The procedure also involves the empirical and theoretical risk premia which – as a by-product – are also estimated. We apply this procedure to data from the German electricity exchange EEX, where we split the empirical analysis into base load and peak load prices. We find an overall negative risk premium for the base load futures contracts, except for contracts close to delivery, where a small positive risk premium is detected. Peak load contracts, on the other hand, show a clear positive risk premium, when they are close to delivery, while contracts in the longer end also have a negative premium.

Fractional lower order moment based adaptive algorithms for active noise control of impulsive noise sources

This letter deals with active noise control (ANC) for impulsive noise sources being modeled using non-Gaussian stable process. The filtered-x least mean square algorithm is based on minimization of the variance of the error signal and becomes unstable for impulsive noise. The filtered-x least mean p-power algorithm-based on minimizing the fractional lower order moment-gives a robust performance for impulsive ANC; however, its convergence speed is very slow. This letter proposes modifying and employing a generalized normalized LMP algorithm for impulsive ANC. Extensive simulations are carried out which demonstrate the effectiveness of the proposed algorithm.

Function-indexed empirical processes based on an infinite source Poisson transmission stream

We study the asymptotic behavior of empirical processes generated by measurable bounded functions of an infinite source Poisson transmission process when the session length have infinite variance. In spite of the boundedness of the function, the normalized fluctuations of such an empirical process converge to a non-Gaussian stable process. This phenomenon can be viewed as caused by the long-range dependence in the transmission process. Completing previous results on the empirical mean of similar types of processes, our results on non-linear bounded functions exhibit the influence of the limit transmission rate distribution at high session lengths on the asymptotic behavior of the empirical process. As an illustration, we apply the main result to estimation of the distribution function of the steady state value of the transmission process.

On family of fractional lower order moment (FLOM)-based algorithms for active noise control of impulsive noise sources

Active noise control (ANC) is based on the principle of destructive interference of propagating acoustic waves; essentially a cancelling signal is generated and combined with the primary noise to achieve acoustic cancellation around location of the error microphone. In this paper, we consider a very challenging application of ANC for impulsive noise. The impulsive noise can be modeled using non-Gaussian stable process for which second order moment does not exist. The most famous filtered-x-LMS (FxLMS) algorithm for ANC systems, is based on minimization of the variance of the error signal, and therefore, becomes unstable for the impulsive noise. It has been shown that filtered-x least mean p-power (FxLMP) algorithm; based on minimizing the fractional lower order moment (FLOM) that does exist for stable distributions; gives robust performance for impulsive ANC. However, the convergence speed of the FxLMP algorithm is very slow. Recently; we have proposed various variants of FxLMP algorithm, so that an improved convergence and noise reduction performance is achieved. In this paper, we propose modifying and employing generalized normalized LMP algorithm (GNLMP) algorithm for ANC of impulsive noise. The computational complexity of proposed algorithm is comparable to the existing FLOM-based ANC algorithms. Extensive simulations are carried out, which demonstrate the effectiveness of proposed algorithm. We observe that, in comparison with the existing FLOM-based ANC algorithms, the proposed algorithm gives best performance for ANC of impulsive noise sources.

Improving robustness of filtered- x least mean p-power algorithm for active attenuation of standard symmetric- α-stable impulsive noise

The paper concerns active control of impulsive noise having peaky distribution with heavy tail. Such impulsive noise can be modeled using non-Gaussian stable process for which second order moments do not exist. The most famous filtered- x least mean square (FxLMS) algorithm for active noise control (ANC) systems is based on the minimization of variance (second order moment) of error signal, and hence, becomes unstable for the impulsive noise. In order to improve the robustness of adaptive algorithms for processes having distributions with heavy tails (i.e. signals with outliers), either (1) a robust optimization criterion may be used to derive the adaptive algorithm or (2) the large amplitude samples may be ignored or replaced by an appropriate threshold value. Among the existing algorithms for ANC of impulsive noise, one is based on the minimizing least mean p-power (LMP) of the error signal, resulting in FxLMP algorithm (approach 1). The other is based on modifying; on the basis of statistical properties; the reference signal in the update equation of the FxLMS algorithm (approach 2). In this paper we propose two solutions to improve the robustness of the FxLMP algorithm. In first proposed algorithm, the reference and the error signals are thresholded before being used in the update equation of FxLMP algorithm. As another solution to improve the performance of FxLMP algorithm, a modified normalized step size is proposed. The computer simulations are carried out, which demonstrate the effectiveness of the proposed algorithms.

Improving performance of FxLMS algorithm for active noise control of impulsive noise

This paper concerns active noise control (ANC) of impulsive noise modeled using non-Gaussian stable processes. The most famous filtered-x least mean square (FxLMS) algorithm for ANC systems is based on minimization of variance of error signal. For the impulse noise, the FxLMS algorithm becomes unstable, as second-order moments do not exist for non-Gaussian processes. Among the existing algorithms for ANC of impulsive noise, one is based on minimizing least mean p-power (LMP) of the error signal, resulting in FxLMP algorithm. The other is based on modifying, on the basis of statistical properties, the reference signal in update of FxLMS algorithm. In this paper, the proposed algorithm is an extension of the later approach. Extensive simulations are carried out, which demonstrate the effectiveness of the proposed algorithm. It achieves the best performance among the existing algorithms, and at the same computational complexity as that of FxLMS algorithm.

Fractional theory for transport in disordered semiconductors

The article is devoted to theoretical description of charge carrier transport in disordered semiconductors. The main idea of the approach lies in the use of fractional calculus. The physical reasons of introducing fractional derivatives in semiconductor theory are discussed, the process of derivation of fractional differential equations is demonstrated, the tied link of their solutions with non-Gaussian stable processes is shown. The last section of the article contains solutions of some concrete problems: multiple trapping, transient photocurrent and drift mobility, dispersive transport percolation model of semiconductors, transport in bilayer semiconductor and so on. Some numerical results are obtained and their agreement with experimental data is demonstrated.

On the Structure of Stationary Stable Processes

A connection between structural studies of stationary non-Gaussian stable processes and the ergodic theory of nonsingular flows is established and exploited. Using this connection, a unique decomposition of a stationary stable process into three independent stationary parts is obtained. It is shown that the dissipative part of a flow generates a mixed moving average part of a stationary stable process, while the identity part of a flow essentially gives the harmonizable part. The third part of a stationary process is determined by a conservative flow without fixed points and by a related cocycle.

Adaptive filtering for non-Gaussian stable processes

A large class of physical phenomena observed in practice exhibit non-Gaussian behavior. In the letter /spl alpha/-stable distributions, which have heavier tails than Gaussian distributions, are considered to model non-Gaussian signals. Adaptive signal processing in the presence of such a noise is a requirement of many practical problems. Since direct application of commonly used adaptation techniques fail in these applications, new algorithms for adaptive filtering for /spl alpha/-stable random processes are introduced.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

Asymptotic dependence of moving average type self-similar stable random Fields

As non-Gaussian stable stochastic processes have infinite second moments, one cannot use the covariance function to describe their dependence structure.

On the spectral representation of symmetric stable processes

The so-called spectral representation theorem for stable processes linearly imbeds each symmetric stable process of index p into L p (0 < p ≤ 2). We use the theory of L p isometries for 0 < p < 2 to study the uniqueness of this representation for the non-Gaussian stable processes. We also determine the form of this representation for stationary processes and for substable processes. Complex stable processes are defined, and a complex version of the spectral representation theorem is proved. As a corollary to the complex theory we exhibit an imbedding of complex L q into real or complex L p for 0 < p < q ≤ 2.

On the convergence of sequences of stochastic processes

0. Introduction. The purpose of this paper is the formulation and investigation of some convergence concepts for sequences of stochastic processes {xn(t, c), tG [0, 1]}. A related result of these investigations appears as a generalization of the central-limit problem for sequences of sums of independent random variables, embodied in the discussions in ??3 and 5 of the sequence (A); Gnedenko's necessary and sufficient conditions for the convergence in distribution of such sums are used and extended. ?4 contains a version of the Helly-Bray theorem (Theorem 9, Corollary) on probability spaces. The tools used are those developed in [2; 7; 8], and [9] for some special cases of convergence of stable processes. The methods of ?4 are a straightforward adaptation of those of [2]; the convergence property of F [ I used in Theorem 9 was found to be necessary by Udagawa (in the publications of the Union of Japanese Scientists and Engineers) in considering the case of sequences (A) of normed-sum type converging to non-Gaussian stable processes. The processes are throughout this paper assumed to have independent increments. Some special cases of this convergence problem are considered in [1; 3; and 4] without this condition; no general results seem to be known at present. 1. Notation and definitions. Let (Q, 63, p) denote a probability space. A real stochastic process defined on (Q, 63, p) with real parameter set T we shall denote by { x(t, co), tE T}. We shall consider sequences of stochastic processes Xn(t, w), tET}, converging in some sense to a stochastic process {x(t, w), tET}. Let