Operator Stable Law Research Articles

On multivariate fractional random fields: Tempering and operator-stable laws

In this paper, we define a new and broad family of vector-valued random fields called tempered operator fractional operator-stable random fields (TRF, for short). TRF is typically non-Gaussian and generalizes tempered fractional stable stochastic processes. TRF comprises moving average and harmonizable-type subclasses that are constructed by tempering (matrix-) homogeneous, matrix-valued kernels in time- and Fourier-domain stochastic integrals with respect to vector-valued, strictly operator-stable random measures. We establish the existence and fundamental properties of TRF. Assuming both Gaussianity and isotropy, we show the equivalence between certain moving average and harmonizable subclasses of TRF. In addition, we establish sample path properties in the scalar-valued case for several Gaussian instances.

Large deviations for stable like random walks on $\mathbb Z^d$ with applications to random walks on wreath products

We derive Donsker-Vardhan type results for functionals of the occupation times when the underlying random walk on $\mathbb Z^d$ is in the domain of attraction of an operator stable law on $\mathbb R^d$. Applications to random walks on wreath products (also known as lamplighter groups) are given.

A bivariate infinitely divisible distribution with exponential and Mittag–Leffler marginals

We introduce a bivariate distribution supported on the first quadrant with exponential, and heavy tailed Mittag–Leffer, marginal distributions. Although this distribution belongs to the class of geometric operator stable laws, it is a rather special case that does not follow their general theory. Our results include the joint density and distribution function, Laplace transform, conditional distributions, joint moments, and tail behavior. We also establish infinite divisibility and stability properties of this model, and clarify its connections with operator stable and geometric operator stable laws.

Moment of levy measure of operator-stable law

In 1979, Jurek gave a characterization of the moment of a full operator-stable μ by eigenvalues of exponent matrix of μ. Here, a characterization of the moment of Levy measure (restricted on a neighbor of 0) of a full operator-stable μ by eigenvalues of exponent matrix of μ is given.

Limit theorems for continuous-time random walks with infinite mean waiting times

A continuous-time random walk is a simple random walk subordinated to a renewal process used in physics to model anomalous diffusion. In this paper we show that, when the time between renewals has infinite mean, the scaling limit is an operator Lévy motion subordinated to the hitting time process of a classical stable subordinator. Density functions for the limit process solve a fractional Cauchy problem, the generalization of a fractional partial differential equation for Hamiltonian chaos. We also establish a functional limit theorem for random walks with jumps in the strict generalized domain of attraction of a full operator stable law, which is of some independent interest.

Limit theorems for continuous-time random walks with infinite mean waiting times

A continuous-time random walk is a simple random walk subordinated to a renewal process used in physics to model anomalous diffusion. In this paper we show that, when the time between renewals has infinite mean, the scaling limit is an operator Lévy motion subordinated to the hitting time process of a classical stable subordinator. Density functions for the limit process solve a fractional Cauchy problem, the generalization of a fractional partial differential equation for Hamiltonian chaos. We also establish a functional limit theorem for random walks with jumps in the strict generalized domain of attraction of a full operator stable law, which is of some independent interest.

Operator geometric stable laws

Operator geometric stable laws are the weak limits of operator normed and centered geometric random sums of independent, identically distributed random vectors. They generalize operator stable laws and geometric stable laws. In this work we characterize operator geometric stable distributions, their divisibility and domains of attraction, and present their application to finance. Operator geometric stable laws are useful for modeling financial portfolios where the cumulative price change vectors are sums of a random number of small random shocks with heavy tails, and each component has a different tail index.

Moving Averages of Random Vectors with Regularly Varying Tails

Regular variation is an analytic condition on the tails of a probability distribution which is necessary for an extended central limit theorem to hold, when the tails are too heavy to allow attraction to a normal limit. The limiting distributions which can occur are called operator stable. In this paper we show that moving averages of random vectors with regularly varying tails are in the generalized domain of attraction of an operator stable law. We also prove that the sample autocovariance matrix of these moving averages is in the generalized domain of attraction of an operator stable law on the vector space of symmetric matrices.AMS 1990 subject classification. Primary 62M10, secondary 62E20, 62F12, 60F05.

One-dimensional marginals of operator stable laws and their domains of attraction

Operator stable laws are the weak limits of ane normalized partial sums of i.i.d. random vectors. It is known that the one-dimensional marginals of operator stable laws need not be stable, or even attracted to a stable law. In this paper we show that for any operator stable law, there exists a basis in which the marginals along every coordinate axis are attracted to a stable or semistable law. This connection between operator stable and semistable laws is new and surpris- ing. We also characterize those operator stable laws whose marginals are stable or semistable. Finally we consider the marginals of random vectors attracted to some operator stable law.

On estimation of the spectral measure of certain nonnormal operator stable laws

Abstract For data belonging to the domain of normal attraction of nonnormal operator stable laws we present a strongly consistent estimate of the s pectral measure. The cases of a known or unknown exponent are considered.

Moments of Limits of Lightly Trimmed Sums of Random Vectors in the Generalized Domain of Normal Attraction of Non-Gaussian Operator-Stable Laws

Existence and nonexistence for moments of limiting random vectors of normalized, lightly trimmed sums of random vectors in the generalized domain of normal attraction of non-Gaussian operator-stable laws are studied. The idea of representing the limiting random vectors by infinite series is essentially used in the proofs.

Pseudomoments for generalized domains of attraction

In this paper we resolve a conjecture of Weiner (1987) concerning the existence of the so-called pseudomoments for a random vector which belongs to the generalized domain of attraction of an operator-stable law. The proof is a direct consequence of the spectral decomposition theorem for generalized domains of attraction in Meerschaert (1991).

Moments of Random Vectors Which Belong to Some Domain of Normal Attraction

Let $X$ be a random vector on $\mathbb{R}^k$ whose distribution $\mu$ belongs to the domain of normal attraction of some operator stable law $\nu$. For a given $\nu$ it has been shown elsewhere that for certain ranges of $\alpha$ depending on $\nu$, either $E|\langle X, \theta\rangle|^\alpha$ is finite for every $\theta \neq 0$ or is infinite for every $\theta \neq 0$. In this paper we show that the set of $\alpha$ for which $E|\langle X, \theta\rangle|^\alpha$ exists depends, in general, on both $\theta$ and $\nu$, and we obtain a complete description of the cases in which $E|\langle X, \theta\rangle|^\alpha$ can be guaranteed either to exist or to diverge, just on the basis of $\theta$ and $\nu$.

Operator stable laws: Series representations and domains of normal attraction

LetX,X 1,X 2,... be i.i.d. random vectors in ℝd. The limit laws μ that can arise by suitable affine normalizations of the partial sums,S n=X 1+...+X n, are calledoperator-stable laws. These laws are a natural extension to ℝ d of the stable laws onℝ. Thegeneralized domain of attraction of μ[GDOA(μ)] is comprised of all random vectorsX whose partial sums can be affinely normalized to converge to μ. If the linear part of the affine transformation is restricted to take the formn −B for some exponent operatorB naturally associated to μ thenX is in thegeneralized domain of normal attraction of μ [GDONA(μ)]. This paper extends the theory of operator-stable laws μ and their domains of attraction and normal attraction.

Moments of distributions attracted to operator-stable laws

Bounds on the norming operators for distributions in the domain of attraction of an operator-stable distribution are found. These bounds are used to establish the existence and nonexistence of moments of distributions in the domain of attraction of an operator-stable distribution. Similar results for stochastically compact sequences are obtained.

Operator-Semistable Distributions on ${\bf R}^d$

Previous article Next article Operator-Semistable Distributions on ${\bf R}^d$V. ChornyV. Chornyhttps://doi.org/10.1137/1131095PDFBibTexSections ToolsAdd to favoritesExport CitationTrack CitationsEmail SectionsAbout[1] Peter Lancaster, Theory of matrices, Academic Press, New York, 1969xii+316 39:6885 0186.05301 Google Scholar[2] V. M. Kruglov, On an extension of the class of stable distributions, Theory Probab. Appl., 17 (1972), 685–694 10.1137/1117081 0279.60035 LinkGoogle Scholar[3] L. S. Pontryagin, Ordinary Differential Equations, Nauka, Moscow, 1974, (In Russian.) Google Scholar[4] R. Jajte, Semi-stable probability measures on ${\bf R}\sp{N}$, Studia Math., 61 (1977), 29–39 55:9210 0365.60017 CrossrefGoogle Scholar[5] William N. Hudson and , J. David Mason, Operator-stable laws, J. Multivariate Anal., 11 (1981), 434–447 10.1016/0047-259X(81)90086-5 83a:60007 0466.60016 CrossrefGoogle Scholar[6] A. Luczak, Operator semistable probability measures on ${\bf R}\sp{N}$, Colloq. Math., 45 (1981), 287–300 83k:60027 0501.60022 Google Scholar[7] M. Yamazato, OL distributions on Euclidean spaces, Theory Probab. Appl., 29 (1984), 1–17 10.1137/1129001 0555.60015 LinkGoogle Scholar Previous article Next article FiguresRelatedReferencesCited ByDetails Multiple Points of Operator Semistable Lévy ProcessesJournal of Theoretical Probability, Vol. 33, No. 1 | 14 September 2018 Cross Ref Asymptotic Behavior of Semistable Lévy Exponents and Applications to Fractal Path PropertiesJournal of Theoretical Probability, Vol. 31, No. 1 | 4 November 2016 Cross Ref The Hausdorff Dimension of Operator Semistable Lévy ProcessesJournal of Theoretical Probability, Vol. 27, No. 2 | 12 May 2012 Cross Ref Almost sure limit theorems of mantissa type for semistable domains of attractionActa Mathematica Hungarica, Vol. 114, No. 4 | 1 Mar 2007 Cross Ref Some new limit theorems for vector space- and group-valued random variablesJournal of Mathematical Sciences, Vol. 93, No. 4 | 1 Feb 1999 Cross Ref Series representation for operator semistable laws and domains of normal attractionJournal of Mathematical Sciences, Vol. 92, No. 4 | 1 Dec 1998 Cross Ref The domain of normal attraction of a semistable distribution on a semidirect product compact group and R dJournal of Mathematical Sciences, Vol. 76, No. 1 | 1 Aug 1995 Cross Ref Recurrence and transience of operator semi-stable processesProceedings of the Japan Academy, Series A, Mathematical Sciences, Vol. 71, No. 5 | 1 Jan 1995 Cross Ref Volume 31, Issue 4| 1987Theory of Probability & Its Applications563-742 History Submitted:18 February 1985Published online:17 July 2006 InformationCopyright © 1987 Society for Industrial and Applied MathematicsPDF Download Article & Publication DataArticle DOI:10.1137/1131095Article page range:pp. 703-705ISSN (print):0040-585XISSN (online):1095-7219Publisher:Society for Industrial and Applied Mathematics

On the Existence and Convergence of Pseudomoments for Variables in the Domain of Normal Attraction of an Operator Stable Distribution

Integrals are constructed to replace absolute moments for variables in the domain of normal attraction of an operator stable law. These integrals, called pseudomoments, improve on the geometric information contained in absolute moments. Existence and convergence to appropriate values of these integrals are shown for the variables and their affine normalized sums.

On the existence and convergence of pseudomoments for variables in the domain of normal attraction of an operator stable distribution

Integrals are constructed to replace absolute moments for variables in the domain of normal attraction of an operator stable law. These integrals, called pseudomoments, improve on the geometric information contained in absolute moments. Existence and convergence to appropriate values of these integrals are shown for the variables and their affine normalized sums.

Supplements to operator-stable and operator-semistable laws on Euclidean spaces

Operator-stable laws and operator-semistable laws (introduced as limit distributions by M. Sharpe and R. Jajte, respectively) are characterized by decomposability properties. Disintegration of their corresponding Lévy measures requires appropriate cross sections. Furthermore in both situations the Lévy measures constitute a Bauer simplex whose extreme boundary can be explicitly given. Finally the infinitely differentiable Lebesgue density of an operator-semistable law is shown to be even analytic in some cases.

Domains of attraction of nonnormal operator-stable laws

A sequence of independent, identically distributed random vectors X 1, X 2, X 3,… is said to belong to the domain of attraction of a random vector Y is there exist linear operators A n and constant vectors b n such that A n ( X 1,…, X n )+ b n converges in distribution to Y. We present a simple, necessary, and sufficient condition for the existence of such A n , B n in the case where Y has no normal component.