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 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 Upgrading Probability via Fractions of Events

2016 ◽  
Vol 24 (1) ◽  
pp. 29-41 ◽  
Author(s):  
Roman Frič ◽  
Martin Papčo
Keyword(s):  
Probability Theory ◽  
Random Variables ◽  
Random Variable ◽  
Classical Probability ◽  
Classical Probability Theory ◽  
Black And White ◽  
Random Events ◽  
Indicator Functions ◽  
Quantum Phenomena ◽  
Special Case

Abstract The influence of “Grundbegriffe” by A. N. Kolmogorov (published in 1933) on education in the area of probability and its impact on research in stochastics cannot be overestimated. We would like to point out three aspects of the classical probability theory “calling for” an upgrade: (i) classical random events are black-and-white (Boolean); (ii) classical random variables do not model quantum phenomena; (iii) basic maps (probability measures and observables { dual maps to random variables) have very different “mathematical nature”. Accordingly, we propose an upgraded probability theory based on Łukasiewicz operations (multivalued logic) on events, elementary category theory, and covering the classical probability theory as a special case. The upgrade can be compared to replacing calculations with integers by calculations with rational (and real) numbers. Namely, to avoid the three objections, we embed the classical (Boolean) random events (represented by the f0; 1g-valued indicator functions of sets) into upgraded random events (represented by measurable {0; 1}-valued functions), the minimal domain of probability containing “fractions” of classical random events, and we upgrade the notions of probability measure and random variable.

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.

Parameter Estimation

2011 ◽  
pp. 172-180
Author(s):  
Karl-Ernst Biebler
Keyword(s):  
Parameter Estimation ◽  
Maximum Likelihood ◽  
Random Variables ◽  
Random Variable ◽  
Likelihood Method ◽  
Approximate Methods ◽  
Normal Distributions ◽  
Parameter Estimations ◽  
Binomial Distributions ◽  
Starting Point

Parameters are numbers which characterize random variables. They make possible the summarizing description of the observations, serve as the basis of statistical decisions and are calculated from the data. Point estimations and confidence estimations are introduced. Samples of the observed random variable are a starting point. The maximum-likelihood method for the construction of parameter estimations is introduced here. Examples concern the normal distributions and the binomial distributions. Approximate methods of the parameter estimation also can be too inaccurate at large sample sizes. This is demonstrated in an example from genetics.

A CHARACTERIZATION OF PROBABILITY MEASURES IN TERMS OF WICK PRODUCT INEQUALITIES

2008 ◽  
Vol 11 (03) ◽  
pp. 377-391 ◽  
Author(s):  
AUREL I. STAN
Keyword(s):  
Random Variable ◽  
Wick Product ◽  
Probability Measures ◽  
Polynomial Functions ◽  
Finite Moments ◽  
Normally Distributed

It is known that if X is a normally distributed random variable, and ♢ and E denote the Wick product and expectation, respectively, then for any non-negative integers m and n, and any polynomial functions f and g of degrees at most m and n, respectively, the following inequality holds: [Formula: see text] We show that this result can be extended to a random variable X, not necessary Gaussian, having an infinite support and finite moments of all orders, if and only if its Szegö–Jacobi sequence {ωk}k ≥ 1 is super-additive.

scholarly journals λ-Analogues of Stirling polynomials of the first kind and their applications

2019 ◽  
Vol 2019 (1) ◽  
Author(s):  
Taekyun Kim ◽  
Dae San Kim ◽  
Han Young Kim ◽  
Sung-Soo Pyo
Keyword(s):  
Recurrence Relations ◽  
Random Variable ◽  
Stirling Numbers ◽  
Mass Function ◽  
Probability Mass Function ◽  
Discrete Random Variable ◽  
Special Polynomials ◽  
Binomial Distributions ◽  
Natural Extensions ◽  
Probability Mass

AbstractRecently, λ-analogues of Stirling numbers of the first kind were studied. In this paper, we introduce, as natural extensions of these numbers, λ-Stirling polynomials of the first kind and r-truncated λ-Stirling polynomials of the first kind. We give recurrence relations, explicit expressions, some identities, and connections with other special polynomials for those polynomials. Further, as applications, we show that both of them appear in an expression of the probability mass function of a suitable discrete random variable, constructed from λ-logarithmic and negative λ-binomial distributions.

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