Convolutions Of Probability Measures Research Articles

GENERALIZED BESSEL AND FRAME MEASURES

Considering a finite Borel measure $ \mu $ on $ \mathbb{R}^d $, a pair of conjugate exponents $ p, q $, and a compatible semi-inner product on $ L^p(\mu) $, we have introduced $ (p,q) $-Bessel and $ (p,q) $-frame measures as a generalization of the concepts of Bessel and frame measures. In addition, we have defined the notions of $ q $-Bessel sequence and $ q$-frame in the semi-inner product space $ L^p(\mu) $. Every finite Borel measure $\nu$ is a $(p,q)$-Bessel measure for a finite measure $ \mu $. We have constructed a large number of examples of finite measures $ \mu $ which admit infinite $ (p,q) $-Bessel measures $ \nu $. We have showed that if $ \nu $ is a $ (p,q) $-Bessel/frame measure for $ \mu $, then $ \nu $ is $ \sigma $-finite and it is not unique. In fact, by using the convolutions of probability measures, one can obtain other $ (p,q) $-Bessel/frame measures for $ \mu $. We have presented a general way of constructing a $ (p,q) $-Bessel/frame measure for a given measure.

Extended Wiener Measure by Nonstandard Analysis for Financial Time Series

We propose a new approach to construct an extended Wiener measure using nonstandard analysis by E. Nelson. For the new definition we construct non-standardized convolution of probability measure for independent random variables. As an application, we consider a simple calculation of financial time series.

Convolution of Probability Measures on Lie Groups and Homogenous Spaces

We study (weakly) continuous convolution semigroups of probability measures on a Lie group G or a homogeneous space G/K, where K is a compact subgroup. We show that such a convolution semigroup is the convolution product of its initial measure and a continuous convolution semigroup with initial measure at the identity of G or the origin of G/K. We will also obtain an extension of Dani-McCrudden's result on embedding an infinitely divisible probability measure in a continuous convolution semigroup on a Lie group to a homogeneous space.

On convolutions on configuration spaces. II. spaces of locally finite configurations

We consider the convolution of probability measures on spaces of locally finite configurations (subsets of a phase space) and their connection with the convolution of the corresponding correlation measures and functionals. In particular, the convolution of Gibbs measures is studied. We also describe a relationship between invariant measures with respect to some operator and properties of the corresponding image of this operator on correlation functions.

A continuous semigroup of notions of independence between the classical and the free one

In this paper, we investigate a continuous family of notions of independence which interpolates between the classical and free ones for noncommutative random variables. These notions are related to the liberation process introduced by Voiculescu. To each notion of independence correspond new convolutions of probability measures, for which we establish formulae and of which we compute simple examples. We prove that there exists no reasonable analogue of classical and free cumulants associated to these notions of independence.

A REMARK ON MONOTONIC CONVOLUTION

Muraki has shown that monotonic convolution of probability measures corresponds with composition of the reciprocal Cauchy transforms corresponding with these measures. We provide a new proof of this fact, not involving the combinatorics of moments.

On the Law of Multiplication of Random Matrices

We recover Voiculescu's results on multiplicative free convolutions of probability measures by different techniques which were already developed by Pastur and Vasilchuk for the law of addition of random matrices. Namely, we study the normalized eigenvalue counting measure of the product of two n×n unitary matrices and the measure of the product of three n×n Hermitian (or real symmetric) positive matrices rotated independently by random unitary (or orthogonal) Haar distributed matrices. We establish the convergence in probability as n→∞ to a limiting nonrandom measure and obtain functional equations for the Herglotz and Stieltjes transforms of that limiting measure.

On the rate of strong convergence for convolutions

An estimate of the rate of strong convergence of convolutions of probability measures is given under the assumption that the components of each convolution converge weakly and in total variation, respectively.

Convolutions of probability measures on semigroups

Convolutions of probability measures on semigroups

Convergence and Convolutions of Probability Measures on a Topological Group

A new technique is developed for studying the convergence of nets of probability measures on a topological group. It is applied to results concerned with the interplay between convergence and convolutions of measures like properties of the convolution mapping, divisibility of measures and convolution semigroups. Our method gives a unified and simple approach to these results.

An Approximation Theorem for Convolutions of Probability Measures

An extension of the usual problem of bounding the total variation of the difference of two probability measures is considered for certain convolutions of probability measures on a measurable Abelian group. The result is a fairly general approximation theorem which also yields an $L_p$ approximation theorem and a large deviation result in some special cases. A limit theorem in equally general setting is proved as a consequence of the main theorem. As the convolutions of probability measures under consideration reduce to the Poisson binomial distribution as a special case, an alternative proof of the approximation theorem in this special case is discussed.