scholarly journals Cauchy–Stieltjes families with polynomial variance functions and generalized orthogonality

2019 ◽  
Vol 39 (2) ◽  
pp. 237-258 ◽  
Author(s):  
Włodzimierz Bryc ◽  
Raouf Fakhfakh ◽  
Wojciech Młotkowski
Keyword(s):  
Free Probability ◽  
Probability Measures ◽  
Arbitrary Degree ◽  
Classical Probability ◽  
Compactly Supported ◽  
Variance Functions ◽  
Additional Variance

This paper studies variance functions of Cauchy–Stieltjes Kernel CSK families generated by compactly supported centered probability measures. We describe several operations that allow us to construct additional variance functions from known ones. We construct a class of examples which exhausts all cubic variance functions, and provide examples of polynomial variance functions of arbitrary degree. We also relate CSK families with polynomial variance functions to generalized orthogonality.Our main results are stated solely in terms of classical probability; some proofs rely on analytic machinery of free probability.

scholarly journals Dual Lukacs regressions of negative orders for noncommutative variables

2014 ◽  
Vol 17 (03) ◽  
pp. 1450021 ◽  
Author(s):  
Kamil Szpojankowski
Keyword(s):  
Free Probability ◽  
Random Variable ◽  
Probability Measures ◽  
Classical Probability ◽  
Binomial Distributions ◽  
Beta Distributions

In the paper we study characterizations of probability measures in free probability. By constancy of regressions for random variable 𝕍1/2(𝕀 - 𝕌)𝕍1/2 given by 𝕍1/2𝕌𝕍1/2, where 𝕌 and 𝕍 are free, we characterize free Poisson and free binomial distributions. Our paper is a free probability analogue of results known in classical probability,3 where gamma and beta distributions are characterized by constancy of 𝔼((V(1 - U))i|UV), for i ∈ {-2, -1, 1, 2}. This paper together with previous results18 exhaust all cases of characterizations from Ref. 3.

scholarly journals On the Outlying Eigenvalues of a Polynomial in Large Independent Random Matrices

2019 ◽  
Author(s):  
Serban T Belinschi ◽  
Hari Bercovici ◽  
Mireille Capitaine
Keyword(s):  
Probability Theory ◽  
Random Matrices ◽  
Empirical Distribution ◽  
Free Probability ◽  
Probability Measures ◽  
Unitary Operators ◽  
Free Probability Theory ◽  
Wigner Matrix ◽  
Asymptotic Eigenvalue ◽  
Special Case

Abstract Given a selfadjoint polynomial $P(X,Y)$ in two noncommuting selfadjoint indeterminates, we investigate the asymptotic eigenvalue behavior of the random matrix $P(A_N,B_N)$, where $A_N$ and $B_N$ are independent Hermitian random matrices and the distribution of $B_N$ is invariant under conjugation by unitary operators. We assume that the empirical eigenvalue distributions of $A_N$ and $B_N$ converge almost surely to deterministic probability measures $\mu$ and $\nu$, respectively. In addition, the eigenvalues of $A_N$ and $B_N$ are assumed to converge uniformly almost surely to the support of $\mu$ and $\nu ,$ respectively, except for a fixed finite number of fixed eigenvalues (spikes) of $A_N$. It is known that almost surely the empirical distribution of the eigenvalues of $P(A_N,B_N)$ converges to a certain deterministic probability measure $\eta \ (\textrm{sometimes denoted}\ P^\square(\mu,\nu))$ and, when there are no spikes, the eigenvalues of $P(A_N,B_N)$ converge uniformly almost surely to the support of $\eta$. When spikes are present, we show that the eigenvalues of $P(A_N,B_N)$ still converge uniformly to the support of $\eta$, with the possible exception of certain isolated outliers whose location can be determined in terms of $\mu ,\nu ,P$, and the spikes of $A_N$. We establish a similar result when $B_N$ is replaced by a Wigner matrix. The relation between outliers and spikes is described using the operator-valued subordination functions of free probability theory. These results extend known facts from the special case in which $P(X,Y)=X+Y$.

ON STOCHASTIC DOMINANCE OF NILPOTENT OPERATORS

2013 ◽  
Vol 16 (01) ◽  
pp. 1350009 ◽  
Author(s):  
SYEDA RABAB MUDAKKAR ◽  
SERGEY UTEV
Keyword(s):  
Shift Operator ◽  
Free Probability ◽  
Stochastic Order ◽  
London Mathematical Society ◽  
Probability Measures ◽  
Cambridge University ◽  
Renewal Equations ◽  
Lecture Note Series ◽  
Nilpotent Operators ◽  
Meixner Class

In this paper, motivated by Nica and Speicher [Lectures on the Combinatorics of Free Probability, London Mathematical Society Lecture Note Series, Vol. 335 (Cambridge University Press, 2006)] and Kubo and Kuo [MRM-factors for the probability measures in the Meixner class, Infin. Dimens. Anal. Quantum Probab. Relat. Top.13 (2010) 525–550], we characterize a particular nilpotent case of a truncated forward shift operator by applying the technique of the random walks with repeated reflections and associated renewal equations. We also establish a stochastic order relationship by applying the crossing criteria.

scholarly journals Analytic function theory for operator-valued free probability

2017 ◽  
Vol 2017 (729) ◽  
Author(s):  
John D. Williams
Keyword(s):  
Probability Theory ◽  
Complex Analysis ◽  
Function Theory ◽  
Free Probability ◽  
Asymptotic Properties ◽  
Cauchy Transform ◽  
Probability Measures ◽  
Cauchy Transforms ◽  
Upper Half Plane ◽  
Object Of Study

AbstractIt is a classical result in complex analysis that the class of functions that arise as the Cauchy transform of probability measures may be characterized entirely in terms of their analytic and asymptotic properties. Such transforms are a main object of study in non-commutative probability theory as the function theory encodes information on the probability measures and the various convolution operations. In extending this theory to operator-valued free probability theory, the analogue of the Cauchy transform is a non-commutative function with domain equal to the non-commutative upper-half plane. In this paper, we prove an analogous characterization of the Cauchy transforms, again, entirely in terms of their analytic and asymptotic behavior. We further characterize those functions which arise as the Voiculescu transform of

scholarly journals Real Functions and the Extension of Generalized Probability Measures

2013 ◽  
Vol 55 (1) ◽  
pp. 85-94
Author(s):  
Jana Havlíčková
Keyword(s):  
Category Theory ◽  
Continuous Functions ◽  
Probability Measures ◽  
Fuzzy Probability ◽  
Classical Probability ◽  
Random Events ◽  
Elementary Methods ◽  
Generalized Probability ◽  
Fuzzy Probability Theory

Abstract In the classical probability, as well as in the fuzzy probability theory, random events and probability measures are modelled by functions into the closed unit interval [0,1]. Using elementary methods of category theory, we present a classification of the extensions of generalized probability measures (probability measures and integrals with respect to probability measures) from a suitable class of generalized random events to a larger class having some additional (algebraic and/or topological) properties. The classification puts into a perspective the classical and some recent constructions related to the extension of sequentially continuous functions.

scholarly journals Positive Definite and Related Functions on Hypergroups

1991 ◽  
Vol 43 (2) ◽  
pp. 242-254 ◽  
Author(s):  
Walter R. Bloom ◽  
Paul Ressel
Keyword(s):  
Positive Definite ◽  
Probability Measures ◽  
Representation Theorems ◽  
Compactly Supported ◽  
Completely Monotone ◽  
Considerable Simplification ◽  
Commutative Hypergroup ◽  
Completely Alternating

AbstractIn this paper we make use of semigroup methods on the space of compactly supported probability measures to obtain a complete Lévy-Khinchin representation for negative definite functions on a commutative hypergroup. In addition we obtain representation theorems for completely monotone and completely alternating functions. The techniques employed here also lead to considerable simplification of the proofs of known results on positive definite and negative definite functions on hypergroups.

A CONNECTION BETWEEN FREE AND CLASSICAL INFINITE DIVISIBILITY

2004 ◽  
Vol 07 (04) ◽  
pp. 573-590 ◽  
Author(s):  
O. E. BARNDORFF-NIELSEN ◽  
S. THORBJØRNSEN
Keyword(s):  
Probability Measure ◽  
Free Probability ◽  
Infinite Divisibility ◽  
Probability Measures ◽  
Stochastic Integration ◽  
Alternative Definition ◽  
Infinitely Divisible ◽  
Definition Of ◽  
Divisible Probability Measure

In this paper we continue our studies, initiated in Refs. 2–4, of the connections between the classes of infinitely divisible probability measures in classical and in free probability. We show that the free cumulant transform of any freely infinitely divisible probability measure equals the classical cumulant transform of a certain classically infinitely divisible probability measure, and we give several characterizations of the latter measure, including an interpretation in terms of stochastic integration. We find, furthermore, an alternative definition of the Bercovici–Pata bijection, which passes directly from the classical to the free cumulant transform, without passing through the Lévy–Khintchine representations (classical and free, respectively).

Sign in / Sign up

Export Citation Format

Share Document