Three Lectures on Free Probability
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The Multivariate Moments Problem: a Practical High Dimensional Density Estimation Method
National Institute for Applied Statistics Research Australia University of Wollongong, Australia Working Paper 11-19 The Multivariate Moments Problem: A Practical High Dimensional Density Estimation Method Bradley Wakefield, Yan-Xia Lin, Rathin Sarathy and Krishnamurty Muralidhar Copyright © 2019 by the National Institute for Applied Statistics Research Australia, UOW. Work in progress, no part of this paper may be reproduced without permission from the Institute. National Institute for Applied Statistics Research Australia, University of Wollongong, Wollongong NSW 2522, Australia Phone +61 2 4221 5435, Fax +61 2 42214998. Email: [email protected] The Multivariate Moments Problem: A Practical High Dimensional Density Estimation Method October 8, 2019 Bradley Wakefield National Institute for Applied Statistics Research Australia School of Mathematics and Applied Statistics University of Wollongong, NSW 2522, Australia Email: [email protected] Yan-Xia Lin National Institute for Applied Statistics Research Australia School of Mathematics and Applied Statistics University of Wollongong, NSW 2522, Australia Email: [email protected] Rathin Sarathy Department of Management Science & Information Systems Oklahoma State University, OK 74078, USA Email: [email protected] Krishnamurty Muralidhar Department of Marketing & Supply Chain Management, Research Director for the Center for Business of Healthcare University of Oklahoma, OK 73019, USA Email: [email protected] 1 Abstract The multivariate moments problem and it's application to the estimation of joint density functions is often considered highly impracticable in modern day analysis. Although many re- sults exist within the framework of this issue, it is often an undesirable method to be used in real world applications due to the computational complexity and intractability of higher order moments. -
THE HAUSDORFF MOMENT PROBLEM REVISITED to What
THE HAUSDORFF MOMENT PROBLEM REVISITED ENRIQUE MIRANDA, GERT DE COOMAN, AND ERIK QUAEGHEBEUR ABSTRACT. We investigate to what extent finitely additive probability measures on the unit interval are determined by their moment sequence. We do this by studying the lower envelope of all finitely additive probability measures with a given moment sequence. Our investigation leads to several elegant expressions for this lower envelope, and it allows us to conclude that the information provided by the moments is equivalent to the one given by the associated lower and upper distribution functions. 1. INTRODUCTION To what extent does a sequence of moments determine a probability measure? This problem has a well-known answer when we are talking about probability measures that are σ-additive. We believe the corresponding problem for probability measures that are only finitely additive has received much less attention. This paper tries to remedy that situation somewhat by studying the particular case of finitely additive probability measures on the real unit interval [0;1] (or equivalently, after an appropriate transformation, on any compact real interval). The question refers to the moment problem. Classically, there are three of these: the Hamburger moment problem for probability measures on R; the Stieltjes moment problem for probability measures on [0;+∞); and the Hausdorff moment problem for probability measures on compact real intervals, which is the one we consider here. Let us first look at the moment problem for σ-additive probability measures. Consider a sequence m of real numbers mk, k ¸ 0. It turns out (see, for instance, [12, Section VII.3] σ σ for an excellent exposition) that there then is a -additive probability measure Pm defined on the Borel sets of [0;1] such that Z k σ x dPm = mk; k ¸ 0; [0;1] if and only if m0 = 1 (normalisation) and the sequence m is completely monotone. -
How Non-Crossing Partitions Occur in Free Probability Theory
How non-crossing partitions occur in free probability theory Alexandru Nica (talk at A.I.M. on January 10, 2005) This is one of the survey talks at the workshop on \Braid groups, clusters, and free probability" at the American Institute of Mathematics, January 10-14, 2005. The goal of the talk is to give a quick and elementary introduction to some combinatorial aspects of free probability, addressed to an audience who is not familiar with free probability but is acquainted to lattices of non-crossing partitions. The (attempted) line of approach is to select a cross-section of relevant material, and increase its rate of absorption by working through a set of exercises (the exercises are embedded in the lecture, and are straightforward applications of the material presented). While this is hopefully an efficient approach, it is quite far from being comprehensive; for a more detailed treatment of the subject, one can check the references [3, 5, 8, 9] listed at the end of the lecture. (Among these references, the one which is closest to the spirit of the present lecture is [3] { but, of course, there is a lot of material from [3] that could not be even mentioned here.) The plan of the lecture is as follows. The first section reviews a minimal amount of general terminology, and then the Sections 2 and 3 present some basic tools used in the combinatorics of free probability { the non-crossing cumulants and the R-transform. The main Section 4 is devoted to explaining how convolution in lattices of non-crossing partitions is intimately related to the multiplication of free random variables. -
Free Probability Theory: Deterministic Equivalents and Combinatorics
Free Probability Theory: Deterministic Equivalents and Combinatorics Dissertation zur Erlangung des Grades des Doktors des Naturwissenschaften der Naturwissenschanftlich-Technischen Fakult¨aten der Universit¨atdes Saarlandes von Carlos Vargas Obieta Saarbr¨ucken 2015 Tag des Kolloquiums: 04.05.2015 Dekan: Prof. Dr. Markus Bläser Prüfungsausschuss: Vorsitzender Prof. Dr. Hannah Markwig Berichterstatter Prof. Dr. Roland Speicher Prof. Dr. Jörg Eschmeier Prof. Dr. Florent Benaych-Georges Akademischer Mitarbeiter Dr. Guillaume Cébron 1 This work is dedicated to all those stubbornly trying to bring education to people who live in remote and vulnerable places. In solidarity with the families of the students of the \Ra´ulIsidro Burgos" Rural Teachers' College of Ayotzinapa: Abel Garc´ıaHern´andez Abelardo V´azquezPeniten Ad´anAbrajan de la Cruz Antonio Santana Maestro Benjam´ınAscencio Bautista Bernardo Flores Alcaraz Carlos Iv´anRam´ırezVillarreal Carlos Lorenzo Hern´andezMu~noz C´esarManuel Gonz´alezHern´andez Christian Alfonso Rodr´ıguezTelumbre Christian Tomas Colon Garnica Cutberto Ortiz Ramos Dorian Gonz´alezParral Emiliano Alen Gaspar de la Cruz. Everardo Rodr´ıguez Bello Felipe Arnulfo Rosas Giovanni Galindes Guerrero Israel Caballero S´anchez Israel Jacinto Lugardo Jes´usJovany Rodr´ıguezTlatempa Jonas Trujillo Gonz´alez Jorge Alvarez´ Nava Jorge An´ıbalCruz Mendoza Jorge Antonio Tizapa Legide~no Jorge Luis Gonz´alezParral Jos´e Angel´ Campos Cantor Jos´e Angel´ Navarrete Gonz´alez Jos´eEduardo Bartolo Tlatempa Jos´eLuis Luna Torres Jhosivani Guerrero de la Cruz Julio C´esarL´opez Patolzin Leonel Castro Abarca Luis Angel´ Abarca Carrillo Luis Angel´ Francisco Arzola Magdaleno Rub´enLauro Villegas Marcial Pablo Baranda Marco Antonio G´omezMolina Mart´ınGetsemany S´anchez Garc´ıa Mauricio Ortega Valerio Miguel Angel´ Hern´andezMart´ınez Miguel Angel´ Mendoza Zacar´ıas Sa´ulBruno Garc´ıa, kidnapped on Sept. -
Free Probability Theory
Chapter 22 Free Probability Theory Roland Speicher Department of Mathematics and Statistics Queen's University Kingston, Ontario K7L 3N6 Canada Abstract Free probability theory was created by Dan Voiculescu around 1985, motivated by his e®orts to understand special classes of von Neumann algebras. His dis- covery in 1991 that also random matrices satisfy asymptotically the freeness relation transformed the theory dramatically. Not only did this yield spectac- ular results about the structure of operator algebras, but it also brought new concepts and tools into the realm of random matrix theory. In the following we will give, mostly from the random matrix point of view, a survey on some of the basic ideas and results of free probability theory. 22.1 Introduction Free probability theory allows one to deal with asymptotic eigenvalue distribu- tions in situations involving several matrices. Let us consider two sequences AN and BN of selfadjoint N £N matrices such that both sequences have an asymp- totic eigenvalue distribution for N ! 1. We are interested in the asymptotic eigenvalue distribution of the sequence f(AN ;BN ) for some non-trivial self- adjoint function f. In general, this will depend on the relation between the eigenspaces of AN and of BN . However, by the concentration of measure phe- nomenon, we expect that for large N this relation between the eigenspaces 2 CHAPTER 22. concentrates on typical or generic positions, and that then the asymptotic eigenvalue distribution of f(AN ;BN ) depends in a deterministic way only on the asymptotic eigenvalue distribution of AN and on the asymptotic eigen- value distribution of BN . -
Kernel Distributionally Robust Optimization: Generalized Duality Theorem and Stochastic Approximation
Kernel Distributionally Robust Optimization: Generalized Duality Theorem and Stochastic Approximation Jia-Jie Zhu Wittawat Jitkrittum Empirical Inference Department Empirical Inference Department Max Planck Institute for Intelligent Systems Max Planck Institute for Intelligent Systems Tübingen, Germany Tübingen, Germany [email protected] Currently at Google Research, NYC, USA [email protected] Moritz Diehl Bernhard Schölkopf Department of Microsystems Engineering Empirical Inference Department & Department of Mathematics Max Planck Institute for Intelligent Systems University of Freiburg Tübingen, Germany Freiburg, Germany [email protected] [email protected] Abstract functional gradient, which apply to general optimization and machine learning tasks. We propose kernel distributionally robust op- timization (Kernel DRO) using insights from 1 INTRODUCTION the robust optimization theory and functional analysis. Our method uses reproducing kernel Imagine a hypothetical scenario in the illustrative figure Hilbert spaces (RKHS) to construct a wide where we want to arrive at a destination while avoiding range of convex ambiguity sets, which can be unknown obstacles. A worst-case robust optimization generalized to sets based on integral probabil- (RO) (Ben-Tal et al., 2009) approach is then to avoid ity metrics and finite-order moment bounds. the entire unsafe area (left, blue). Suppose we have This perspective unifies multiple existing ro- historical locations of the obstacles (right, dots). We bust and stochastic optimization methods. may choose to avoid only the convex polytope that We prove a theorem that generalizes the clas- contains all the samples (pink). This data-driven robust sical duality in the mathematical problem of decision-making idea improves efficiency while retaining moments. -
De Finetti Theorems for Easy Quantum Groups
The Annals of Probability 2012, Vol. 40, No. 1, 401–435 DOI: 10.1214/10-AOP619 c Institute of Mathematical Statistics, 2012 DE FINETTI THEOREMS FOR EASY QUANTUM GROUPS By Teodor Banica1, Stephen Curran and Roland Speicher2 Cergy-Pontoise University, UCLA and Queen’s University We study sequences of noncommutative random variables which are invariant under “quantum transformations” coming from an or- thogonal quantum group satisfying the “easiness” condition axioma- tized in our previous paper. For 10 easy quantum groups, we obtain de Finetti type theorems characterizing the joint distribution of any infinite quantum invariant sequence. In particular, we give a new and unified proof of the classical results of de Finetti and Freedman for the easy groups Sn, On, which is based on the combinatorial theory of cumulants. We also recover the free de Finetti theorem of K¨ostler and Speicher, and the characterization of operator-valued free semi- circular families due to Curran. We consider also finite sequences, and prove an approximation result in the spirit of Diaconis and Freedman. Introduction. In the study of probabilistic symmetries, the classical groups Sn and On play central roles. De Finetti’s fundamental theorem states that an infinite sequence of random variables whose joint distribu- tion is invariant under finite permutations must be conditionally indepen- dent and identically distributed. In [20], Freedman considered sequences of real-valued random variables whose joint distribution is invariant under or- thogonal transformations, and proved that any infinite sequence with this property must form a conditionally independent Gaussian family with mean zero and common variance. Although these results fail for finite sequences, approximation results may still be obtained (see [17, 18]). -
On the Combinatorics of Cumulants
Journal of Combinatorial Theory, Series A 91, 283304 (2000) doi:10.1006Âjcta.1999.3017, available online at http:ÂÂwww.idealibrary.com on On the Combinatorics of Cumulants Gian-Carlo Rota Department of Mathematics, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139 and Jianhong Shen School of Mathematics, University of Minnesota, Minneapolis, Minnesota 55455 E-mail: jhshenÄmath.umn.edu View metadata, citation and similar papers at core.ac.uk brought to you by CORE Communicated by the Managing Editors provided by Elsevier - Publisher Connector Received July 26, 1999 dedicated to the memory of gian-carlo rota We study cumulants by Umbral Calculus. Various formulae expressing cumulants by umbral functions are established. Links to invariant theory, symmetric functions, and binomial sequences are made. 2000 Academic Press 1. INTRODUCTION Cumulants were first defined and studied by Danish scientist T. N. Thiele. He called them semi-invariants. The importance of cumulants comes from the observation that many properties of random variables can be better represented by cumulants than by moments. We refer to Brillinger [3] and Gnedenko and Kolmogorov [4] for further detailed probabilistic aspects on this topic. Given a random variable X with the moment generating function g(t), its nth cumulant Kn is defined as d n Kn= n log g(t). dt } t=0 283 0097-3165Â00 35.00 Copyright 2000 by Academic Press All rights of reproduction in any form reserved. 284 ROTA AND SHEN That is, m K : n tn= g(t)=exp : n tn ,(1) n0 n! \ n1 n! + where, mn is the nth moment of X. Generally, if _ denotes the standard deviation, then 2 2 K1=m1 , K2=m2&m1=_ . -
Combinatorics of Free Probability Theory Roland Speicher
Combinatorics of free probability theory Roland Speicher Queen’s University, Kingston, Ontario Abstract. The following manuscript was written for my lec- tures at the special semester ’Free probability theory and operator spaces’, IHP, Paris, 1999. A complementary series of lectures was given by A. Nica. It is planned to combine both manuscripts into a book A. Nica and R. Speicher: Lectures on the Combinatorics of Free Probability, which will be published by Cambridge University Press Contents Part 1. Basic concepts 5 Chapter 1. Basic concepts of non-commutative probability theory 7 1.1. Non-commutative probability spaces and distributions 7 1.2. Haar unitaries and semicircular elements 10 Chapter 2. Free random variables 17 2.1. Definition and basic properties of freeness 17 2.2. The group algebra of the free product of groups 21 2.3. The full Fock space 23 2.4. Construction of free products 27 Part 2. Cumulants 35 Chapter 3. Free cumulants 37 3.1. Motivation: Free central limit theorem 37 3.2. Non-crossing partitions 46 3.3. Posets and M¨obius inversion 50 3.4. Free cumulants 52 Chapter 4. Fundamental properties of free cumulants 59 4.1. Cumulants with products as entries 59 4.2. Freeness and vanishing of mixed cumulants 66 Chapter 5. Sums and products of free variables 75 5.1. Additive free convolution 75 5.2. Description of the product of free variables 84 5.3. Compression by a free projection 87 5.4. Compression by a free family of matrix units 91 Chapter 6. R-diagonal elements 95 6.1. -
The Truncated Stieltjes Moment Problem Solved by Using Kernel Density Functions
View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by Elsevier - Publisher Connector Journal of Computational and Applied Mathematics 236 (2012) 4193–4213 Contents lists available at SciVerse ScienceDirect Journal of Computational and Applied Mathematics journal homepage: www.elsevier.com/locate/cam The truncated Stieltjes moment problem solved by using kernel density functions P.N. Gavriliadis ∗, G.A. Athanassoulis Department of Naval Architecture and Marine Engineering, National Technical University of Athens, Heroon Polytechniou 9, GR-157 73 Zografos, Athens, Greece article info a b s t r a c t Article history: In this work the problem of the approximate numerical determination of a semi-infinite Received 4 July 2010 supported, continuous probability density function (pdf) from a finite number of its Received in revised form 6 September 2011 moments is addressed. The target space is carefully defined and an approximation theorem is proved, establishing that the set of all convex superpositions of appropriate Kernel MSC: Density Functions (KDFs) is dense in this space. A solution algorithm is provided, based 65R32 on the established approximate representation of the target pdf and the exploitation 65F22 of some theoretical results concerning moment sequence asymptotics. The solution 45Q05 algorithm also permits us to recover the tail behavior of the target pdf and incorporate Keywords: this information in our solution. A parsimonious formulation of the proposed solution Truncated Stieltjes moment problem procedure, based on a novel sequentially adaptive scheme is developed, enabling a very Kernel density function efficient moment data inversion. The whole methodology is fully illustrated by numerical Probability density function examples. -
The Bi-Free Extension of Free Probability
The Bi-Free Extension of Free Probability Dan-Virgil Voiculescu Abstract. Free probability is a non-commutative probability the- ory adapted to variables with the highest degree of non-commutativity. The theory has connections with random matrices, combinatorics and operator algebras. Recently we realized that the theory has an extension to systems with left and right variables, based on a notion of bi-freeness. We provide a look at the development of this new direction. The paper is an expanded version of the plenary lecture at the 10th ISAAC Congress in Macau. 1. Introduction Free probability is now in its early thirties. After such a long time I became aware that the theory has a natural extension to a theory with two kinds of variables: left and right. This is not the same as passing from a theory of modules to a theory of bi-modules, since our left and right variables will not commute in general, a non-commutation which will appear already when we shall take a look at what the Gaussian variables of the theory are. We call the theory with left and right variables bi-free probabil- ity and the independence relation that underlies it is called bi-freeness. This new type of independence does not contradict the theorems of Mu- raki [15] and Speicher [20] about the possible types of independence in non-commutative probability with all the nice properties (\classi- cal", free, Boolean and if we give up symmetry also monotonic and anti-monotonic), the reason being that we play a new game here, by replacing the usual sets of random variables by sets with two types of variables. -
Stieltjes Moment Sequences for Pattern-Avoiding Permutations Alin Bostan, Andrew Elvey-Price, Anthony Guttmann, Jean-Marie Maillard
Stieltjes moment sequences for pattern-avoiding permutations Alin Bostan, Andrew Elvey-Price, Anthony Guttmann, Jean-Marie Maillard To cite this version: Alin Bostan, Andrew Elvey-Price, Anthony Guttmann, Jean-Marie Maillard. Stieltjes moment se- quences for pattern-avoiding permutations. The Electronic Journal of Combinatorics, Open Journal Systems, In press, pp.59. 10.37236/xxxx. hal-02425917v3 HAL Id: hal-02425917 https://hal.archives-ouvertes.fr/hal-02425917v3 Submitted on 17 Oct 2020 HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. Distributed under a Creative Commons Attribution - NoDerivatives| 4.0 International License Stieltjes moment sequences for pattern-avoiding permutations Alin Bostan Andrew Elvey Price Inria and Universit´eParis-Saclay LaBRI, Universit´eBordeaux I 1 rue Honor´ed'Estienne d'Orves 351 cours de la Lib´eration 91120 Palaiseau, France 33405 Talence Cedex, France [email protected] [email protected] Anthony John Guttmann Jean-Marie Maillard School of Mathematics and Statistics LPTMC, CNRS, Sorbonne Universit´e The University of Melbourne 4 Place Jussieu, Tour 23, case 121 Vic. 3010, Australia 75252 Paris Cedex 05, France [email protected] [email protected] Submitted: Feb 28, 2020; Accepted: Sep 16, 2020; Published: TBD © The authors.