Absolute continuity of multivariate distributions of class L

It is shown that every genuinely d-dimensional distribution of class L on R^d is absolutely continuous. This extends the known fact in one dimension to all finite dimensions.     

Absolute continuity of operator-self-decomposable distributions on Rd

It is shown that every genuinely d-dimensional operator-self-decomposable distribution is absolutely continuous.     

A characterization of Lévy probability distribution functions on Euclidean spaces

In 1937, Paul Lévy proved two theorems that characterize one-dimensional distribution functions of class L. In 1972, Urbanik generalized Lévy's first theorem. In this note, we generalize Lévy's second theorem and obtain a new characterization of Lévy probability distribution functions on Euclidean spaces. This result is used to obtain a new characterization of operator stable distribution functions on Euclidean spaces and to show that symmetric Lévy distribution functions on Euclidean spaces need not be symmetric unimodal.     

On α-factors of multivariate infinitely divisible probabilities

Let p be an infinitely divisible n-variate probability having μ for Poisson measure. We give here some sufficient conditions for p to belong to the class $\text{I}{}_{\mathrm{\alpha }}{}^{\mathrm{n}}$ of n-variate probabilities having only infinitely divisible α-factors. These results are interesting since they are concerned with the case when μ is a continuous measure.     

On lévy measure and integrability of the norm in C[0, 1]

We prove that integrability of the norm is the best sufficient condition in terms of integrability of functions of the norm for a positive measure to be a Lévy Measure in C[0, 1].     

On α-symmetric multivariate distributions

A random vector is said to have a 1-symmetric distribution if its characteristic function is of the form φ(|t₁| + … + |t_n|). 1-Symmetric distributions are characterized through representations of the admissible functions φ and through stochastic representations of the radom vectors, and some of their properties are studied.     

Operator-selfdecomposable distributions as limit distributions of processes of Ornstein-Uhlenbeck type

Processes of Ornstein-Uhlenbeck type on $\text{R}$^d are analogues of the Ornstein-Uhlenbeck process on $\text{R}$^d with the Brownian motion part replaced by general processes with homogeneous independent increments. The class of operator-selfdecomposable distributions of Urbanik is characterized as the class of limit distributions of such processes. Continuity of the correspondence is proved. Integro-differential equations for operator-selfdecomposable distributions are established. Examples are given for null recurrence and transience of processes of Ornstein-Uhlenbeck type on $\text{R}$¹.     

Semi-stable probability measures on Hilbert spaces

In this paper we define semi-stable probability measures (laws) on a real separable Hilbert space and are identified as limit laws. We characterize them in terms of their Lévy-Khinchine measure and the exponent 0 < p ≤ 2. Finally we prove that every semi-stable probability measure of exponent p has finite absolute moments of order 0 ≤ α < p.     

On the unimodality of multivariate symmetric distribution functions of class L

A theorem is proved that characterizes multivariate distribution functions of class L. This theorem is used to show that every n-dimensional, symmetric distribution function of class L is unimodal in the sense of Kanter.     

Absolute continuity of operator-self-decomposable distributions on R

It is shown that every genuinely d-dimensional operator-self-decomposable distribution is absolutely continuous.     

Some remarks on multivariate stable distributions

This paper deals with multivariate stable distributions. Press has given an explicit algebraic representation of characteristic functions of such distributions [J. Multivariate Analysis2 (1972), 444–462]. We present counter-examples and correct proofs of some of the statements of Press. The properties of multivariate stable distributions, connected with the spectral measure Γ, present in the expression of the characteristic function, are studied.     

A class of multivariate symmetric stable distributions

Multivariate symmetric stable characteristic functions and their properties, as well as conditions for independence and an analogue of the correlation coefficient in bivariate symmetric stable distributions, are discussed.     

Central limit problem on Lp (p ≥ 2) II. Compactness of infinitely divisible laws

Compactness criterion for a sequence of infinitely divisible laws in terms of their Lévy-Khinchine representations is obtained. As a consequence, analog of classical central limit theorems without the assumption of bounded variance on the triangular arrays are proved.     

Characterization of infinitely divisible multivariate gamma distributions

A particular class of p-dimensional exponential distributions have Laplace transforms |I + VT|^?1, V positive definite or positive semi-definite and T = diagonal (t₁,…, t_p). A characterization is given of when these Laplace transforms are infinitely divisible.     

On measures of the distance of a mixture from its parent distribution

In a previous paper in this Journal, Heyde and Leslie [6] examined moment measures of the distance of a mixture from its parent distribution. They confined their attention to the case where the parent distribution is either normal or exponential, and related the moment measures to the more familiar uniform distance between distributions. In this paper we improve on their results by sharpening one of their inequalities. We then use new techniques to extend their investigation to a larger class of parent distributions.     

On the unimodality of spherically symmetric stable distribution functions

Several theorems are obtained concerning the unimodality of spherically symmetric distribution functions. These theorems are used to show that a class of spherically symmetric infinitely divisible distribution functions that contains the class of spherically symmetric stable distribution functions is unimodal.     

On the finite series expansion of multivariate characteristic functions

Three theorems are obtained that relate the asymptotic behavior of a distribution function with the behavior of its characteristic function at the origin. These theorems generalize one dimensional results that have been obtained by the author and by others.     

Limit distributions for risk processes in case of claim amounts of finite expectation

We prove limit theorems for the distribution of $\text{Z}{}_{\mathrm{l}}\text{if {Z}{}_{\mathrm{l}}\text{:l ?}\text{R}{}_{\mathrm{+}}\text{}}$ is a risk process with claim amounts of finite mean. The results are illustrated by several examples and counterexamples.     

Limit distributions of V- and U-statistics in terms of multiple stochastic Wiener-type integrals

We describe the limit distribution of V- and U-statistics in a new fashion. In the case of V-statistics the limit variable is a multiple stochastic integral with respect to an abstract Brownian bridge G_Q. This extends the pioneer work of Filippova (1961) [8]. In the case of U-statistics we obtain a linear combination of G_Q-integrals with coefficients stemming from Hermite Polynomials. This is an alternative representation of the limit distribution as given by Dynkin and Mandelbaum (1983) [7] or Rubin and Vitale (1980) [13]. It is in total accordance with their results for product kernels.