Characterizations of free Meixner distributions
|
|
- Jordan Andrews
- 5 years ago
- Views:
Transcription
1 Characterizations of free Meixner distributions Texas A&M University March 26, 2010
2 Jacobi parameters. Matrix. β 0 γ β 1 γ m n J =. 0 1 β 2 γ.. 2 ; J n = β A new sequence m 1, m 2, m 3,.... Favard s Theorem. [Stone 1932] All γ i 0 {m i } are moments of a measure, m i = x i dµ(x).
3 Definition. b, c, t R, t, t + c 0. Tridiagonal matrix 0 t b t + c 0 0 J = 0 1 b t + c b t + c µt b,c b µ t b,c = free Meixner distributions. Bernstein-Szegő class: β i, γ i eventually constant.
4 Formula. In terms of the Cauchy transform 1 G µ (z) = z x dµ(x), R dµ(x) = 1 π lim ε 0 + Im G µ(x + iε) dx ( 4(t + c) (x b) 2 ) µ t b,c = 1 2π t + bx + (c/t)x 2 + dx + 0, 1,or 2 atoms.
5 Examples. µ 0,1, µ 0,0, µ 0,1/2.
6 Semicircle law. b, c = 0. µ t 0,0 = 1 4t x 2πt 2. Orthogonal polynomials: Chebyshev polynomials of the second kind U n (x). Generating function U n (x, t)w n = n=0 1 1 xw + tw 2.
7 First characterization. Theorem. [A. 03] Orthogonal polynomials P n (x) for a measure µ have a generating function of the resolvent-type form P n (x)w n 1 = A(w) 1 B(w)x n=0 if and only if µ is a free Meixner distribution.
8 Analogy: Meixner class. Theorem. [Meixner 1934] Orthogonal polynomials P n (x) for a measure µ have a generating function of the exponential form n=0 1 n! P n(x)w n = A(w)e B(w)x if and only if µ is a Meixner distribution: Normal, Gamma, hyperbolic secant. Binomial, Poisson, negative binomial. A lot more interesting and important class. Yet: have an analogy.
9 Lévy process. Random variables {X t : t R}, Increments X t0 X t1 X t2 X t3. X(t 1 ) X(t 0 ), X(t 2 ) X(t 1 ),..., X(t k ) X(t k 1 ) independent and stationary: (X t X s ) X t s µ t s. Example. B t = Brownian motion. µ t = 1 2πt e x2 /t dx.
10 Conditional expectation. E [ ] = expectation. E [ X] = conditional expectation = projection. E [f(x) X] = f(x), E [Y X] = E [Y ] if Y, X independent.
11 Exponential martingale. Denote [ F t (θ) = E e ixtθ] = e ixθ dµ t (x) the Fourier transform. ] E [e ixtθ log Ft(θ) X s = e ixsθ log Fs(θ). t eixtθ log Ft(θ) is Brownian motion a martingale. e ibtθ t θ2 2.
12 Other martingales. t 7 eixt θ log Ft (θ).
13 Proof. For X, Y independent F X+Y (θ) = F X (θ)f Y (θ) F µt (θ) = F t (θ) log F t (θ) = t log F(θ). ] ] E [e ixtθ log Ft(θ) X s = E [e i(xt Xs)θ e ixsθ e t log F(θ) X s [ = E e i(xt Xs)θ] e ixsθ t log F(θ) e = e (t s) log F(θ) e t log F(θ) e ixsθ ixsθ log sf(θ) = e
14 Martingale polynomials. Generating function e xz t log F( iz) = n=0 Each P n = martingale polynomial. 1 n! P n(x, t)z n. Brownian motion: H n (x, t) Hermite polynomials. H n (B t, t) = martingale. Meixner class: orthogonal martingale polynomials.
15 Free independence. Independent: E [f(x)g(y )] = E [f(x)] E [g(y )]. What if X, Y don t commute? What is the correct expression for E [f(x)g(y )h(x)] in terms of X and Y separately? Voiculescu s free independence.
16 Free probability (Voiculescu 1980s). Common structure in Operator Algebras Random Matrices Asymptotic Representation Theory Probability theory with (maximally) non-commuting random variables. Free versions of many probabilistic objects and theorems. (Free) independence, (free) product, (free) infinitely divisible distributions and limit theorems, (free) cumulants, (free) normal, Poisson, binomial distributions, etc.
17 Examples. Free central limit theorem. Limit = semicircular distribution = µ 0,0. Free Poisson limit theorem. Limit = free Poisson distribution = µ 1,0. Binomial distribution = sum of independent coin tosses. Coin toss = projection. Sum of freely independent projections = free binomial distribution = µ t 0, 1, t = number of tosses. In the free case, t 1 real!
18 Processes with free increments. X t = process with freely independent increments. Theorem. [Biane 1998]. {X t } is a Markov processes. Equivalently: is a martingale. t 1 1 X t z + tr(z) Here R(z) = R-transform (Voiculescu), analog of log F(iθ). Free Meixner processes have orthogonal martingale polynomials.
19 Reverse martingales. {X t } a process with freely independent increments. {P n (x, t)} polynomials, P n of degree n. P n a martingale if for s < t, E [P n (X t, t) X s ] = P n (X s, s). P n a reverse martingale if for s < t, E [P n (X s, s) X t ] = f(s) f(t) P n(x t, t).
20 Characterization: reverse martingales. Theorem. [Laha, Lukacs 1960] Their result implies: A Lévy process {X t } has a family of polynomials P n which are both martingales and reverse martingales if and only if each X s has a Meixner distribution. Theorem. [Bożejko, Bryc 06] Their result implies: A free Lévy process {X t } has a family of polynomials P n which are both martingales and reverse martingales if and only if each X s has a free Meixner distribution.
21 Further characterizations. Algebraic Riccati equation (A. 07) Free Jacobi fields (Bożejko, Lytvynov 09) Free quadratic exponential families (Bryc 09) Free quadratic harnesses (Bryc, Wesołowski 05) Etc. Other appearances: Szegö (1922), Bernstein (1930), Boas & Buck (1956), Carlin & McGregor (1957), Geronimus (1961), Allaway (1972), Askey & Ismail (1983), Cohen & Trenholme (1984), Kato (1986), Freeman (1998), Saitoh & Yoshida (2001), Kubo, Kuo & Namli (2006), Belinschi & Nica (2007),...
22 Motivation: Sturm-Liouville operators. Operator Symmetric in L 2 (w(x) dx) for DpD + qd : y (py ) + qy. w w = q p. With appropriate boundary conditions, self-adjoint. Has orthogonal eigenfunctions.
23 Bochner s Theorem. Theorem. [Bochner 1929] DpD + qd has (orthogonal) polynomial eigenfunctions if and only if deg p 2, deg q 1 and w(x) dx = normal distribution (Hermite polynomials) w(x) dx = Gamma distribution (Legendre polynomials) w(x) dx = Beta distribution (Jacobi polynomials) Pearson (1924): for w w = linear quadratic, two more distributions (Bessel polynomials, t-distribution). Easy (calculus).
24 Operator L µ. Replace D with L µ. Definition. For µ a measure, f(x) f(y) L µ [f] = dµ(y) = (I µ) [f]. x y R L µ maps polynomials to polynomials, lowers degree by one. Origin: Maps orthogonal polynomials to associated orthogonal polynomials. Related to the generator of the free Brownian motion.
25 Bochner-Pearson type characterization. Theorem. [A. 09] pl 2 µ + ql µ has polynomial eigenfunctions if and only if µ is a free Meixner distribution.
26 More on free Meixner distributions. The Cauchy transform But also the Hilbert transform G µ (z) = linear quadratic. quadratic dµ(x) = 1 π lim Im G(x + iε) dx. ε 0 + H[µ](x) = 1 π lim Re G(x + iε) dx = linear ε 0 + In fact for H µ = 2πH[µ], H µ = q p. quadratic.
27 Classical-free correspondence. DpD + qd pl 2 µ + ql µ. w w q p H µ. Meaning of H µ : free conjugate variable. Random matrix picture: if then for w(x) dx = e V (x) dx, 1 Z e trv (M) dm µ w w = V = H µ.
28 Random matrix picture. Gaussian Unitary ensemble: e x2 /2 Wigner law (semicircular). Wishart ensemble: x α 1 e x 1 0 x Marchenko-Pastur law (free Poisson). Jacobi ensemble: (1 x) α 1 x β x 1... (free binomial).
29 Summary. w w d + ex a + bx + cx 2 H µ. Correspondence between parameters not precise. normal Poisson binomial semicircular Marchenko-Pastur free binomial normal gamma beta hyperbolic secant gamma negative binomial free h.s. free g. free n.b. Bessel t-distribution?
Characterizations of free Meixner distributions
Characterizations of free Meixner distributions Texas A&M University August 18, 2009 Definition via Jacobi parameters. β, γ, b, c R, 1 + γ, 1 + c 0. Tridiagonal matrix {(β, b, b,...), (1 + γ, 1 + c, 1
More informationFree Meixner distributions and random matrices
Free Meixner distributions and random matrices Michael Anshelevich July 13, 2006 Some common distributions first... 1 Gaussian Negative binomial Gamma Pascal chi-square geometric exponential 1 2πt e x2
More informationMeasures, orthogonal polynomials, and continued fractions. Michael Anshelevich
Measures, orthogonal polynomials, and continued fractions Michael Anshelevich November 7, 2008 MEASURES AND ORTHOGONAL POLYNOMIALS. MEASURES AND ORTHOGONAL POLYNOMIALS. µ a positive measure on R. 2 MEASURES
More informationMeasures, orthogonal polynomials, and continued fractions. Michael Anshelevich
Measures, orthogonal polynomials, and continued fractions Michael Anshelevich November 7, 2008 MEASURES AND ORTHOGONAL POLYNOMIALS. µ a positive measure on R. A linear functional µ[p ] = P (x) dµ(x), µ
More informationMeixner matrix ensembles
Meixner matrix ensembles W lodek Bryc 1 Cincinnati April 12, 2011 1 Based on joint work with Gerard Letac W lodek Bryc (Cincinnati) Meixner matrix ensembles April 12, 2011 1 / 29 Outline of talk Random
More informationFrom random matrices to free groups, through non-crossing partitions. Michael Anshelevich
From random matrices to free groups, through non-crossing partitions Michael Anshelevich March 4, 22 RANDOM MATRICES For each N, A (N), B (N) = independent N N symmetric Gaussian random matrices, i.e.
More informationKernel families of probability measures. Saskatoon, October 21, 2011
Kernel families of probability measures Saskatoon, October 21, 2011 Abstract The talk will compare two families of probability measures: exponential, and Cauchy-Stjelties families. The exponential families
More informationNonlinear Integral Equation Formulation of Orthogonal Polynomials
Nonlinear Integral Equation Formulation of Orthogonal Polynomials Eli Ben-Naim Theory Division, Los Alamos National Laboratory with: Carl Bender (Washington University, St. Louis) C.M. Bender and E. Ben-Naim,
More informationULTRASPHERICAL TYPE GENERATING FUNCTIONS FOR ORTHOGONAL POLYNOMIALS
ULTRASPHERICAL TYPE GENERATING FUNCTIONS FOR ORTHOGONAL POLYNOMIALS arxiv:083666v [mathpr] 8 Jan 009 Abstract We characterize, up to a conjecture, probability distributions of finite all order moments
More informationFree Probability Theory and Random Matrices. Roland Speicher Queen s University Kingston, Canada
Free Probability Theory and Random Matrices Roland Speicher Queen s University Kingston, Canada We are interested in the limiting eigenvalue distribution of N N random matrices for N. Usually, large N
More informationLinearization coefficients for orthogonal polynomials. Michael Anshelevich
Linearization coefficients for orthogonal polynomials Michael Anshelevich February 26, 2003 P n = monic polynomials of degree n = 0, 1,.... {P n } = basis for the polynomials in 1 variable. Linearization
More informationQuadratic harnesses, q-commutations, and orthogonal martingale polynomials. Quadratic harnesses. Previous research. Notation and Motivation
Quadratic harnesses, q-commutations, and orthogonal martingale polynomials Quadratic harnesses Slide 1 W lodek Bryc University of Cincinnati August 2005 Based on joint research with W Matysiak and J Wesolowski
More informationSelfadjoint Polynomials in Independent Random Matrices. Roland Speicher Universität des Saarlandes Saarbrücken
Selfadjoint Polynomials in Independent Random Matrices Roland Speicher Universität des Saarlandes Saarbrücken We are interested in the limiting eigenvalue distribution of an N N random matrix for N. Typical
More informationExponential tail inequalities for eigenvalues of random matrices
Exponential tail inequalities for eigenvalues of random matrices M. Ledoux Institut de Mathématiques de Toulouse, France exponential tail inequalities classical theme in probability and statistics quantify
More informationCOMPUTING MOMENTS OF FREE ADDITIVE CONVOLUTION OF MEASURES
COMPUTING MOMENTS OF FREE ADDITIVE CONVOLUTION OF MEASURES W LODZIMIERZ BRYC Abstract. This short note explains how to use ready-to-use components of symbolic software to convert between the free cumulants
More informationCauchy-Stieltjes kernel families
Cauchy-Stieltjes kernel families Wlodek Bryc Fields Institute, July 23, 2013 NEF versus CSK families The talk will switch between two examples of kernel families K(µ) = {P θ (dx) : θ Θ} NEF versus CSK
More informationHANDBOOK OF LINEAR PARTIAL DIFFERENTIAL EQUATIONS for ENGINEERS and SCIENTISTS
HANDBOOK OF LINEAR PARTIAL DIFFERENTIAL EQUATIONS for ENGINEERS and SCIENTISTS Andrei D. Polyanin Chapman & Hall/CRC Taylor & Francis Group Boca Raton London New York Singapore Foreword Basic Notation
More informationMarkov operators, classical orthogonal polynomial ensembles, and random matrices
Markov operators, classical orthogonal polynomial ensembles, and random matrices M. Ledoux, Institut de Mathématiques de Toulouse, France 5ecm Amsterdam, July 2008 recent study of random matrix and random
More informationSpecial Functions of Mathematical Physics
Arnold F. Nikiforov Vasilii B. Uvarov Special Functions of Mathematical Physics A Unified Introduction with Applications Translated from the Russian by Ralph P. Boas 1988 Birkhäuser Basel Boston Table
More informationIntroduction to orthogonal polynomials. Michael Anshelevich
Introduction to orthogonal polynomials Michael Anshelevich November 6, 2003 µ = probability measure on R with finite moments m n (µ) = R xn dµ(x)
More informationPolynomials in Free Variables
Polynomials in Free Variables Roland Speicher Universität des Saarlandes Saarbrücken joint work with Serban Belinschi, Tobias Mai, and Piotr Sniady Goal: Calculation of Distribution or Brown Measure of
More informationExercises. T 2T. e ita φ(t)dt.
Exercises. Set #. Construct an example of a sequence of probability measures P n on R which converge weakly to a probability measure P but so that the first moments m,n = xdp n do not converge to m = xdp.
More informationMATHEMATICAL FORMULAS AND INTEGRALS
MATHEMATICAL FORMULAS AND INTEGRALS ALAN JEFFREY Department of Engineering Mathematics University of Newcastle upon Tyne Newcastle upon Tyne United Kingdom Academic Press San Diego New York Boston London
More informationTOPICS ON MEIXNER FAMILIES AND ULTRASPHERICAL TYPE GENERATING FUNCTIONS FOR ORTHOGONAL POLYNOMIALS. Marek Bozejko 1 and Nizar Demni 2
TOPICS ON MEIXNE FAMILIES AND ULTASPHEICAL TYPE GENEATING FUNCTIONS FO OTHOGONAL POLYNOMIALS Marek Bozejko and Nizar Demni 2 Abstract. We highlight some connections between different characterizations
More informationTEST CODE: MMA (Objective type) 2015 SYLLABUS
TEST CODE: MMA (Objective type) 2015 SYLLABUS Analytical Reasoning Algebra Arithmetic, geometric and harmonic progression. Continued fractions. Elementary combinatorics: Permutations and combinations,
More informationMATHEMATICAL FORMULAS AND INTEGRALS
HANDBOOK OF MATHEMATICAL FORMULAS AND INTEGRALS Second Edition ALAN JEFFREY Department of Engineering Mathematics University of Newcastle upon Tyne Newcastle upon Tyne United Kingdom ACADEMIC PRESS A Harcourt
More informationLimit Laws for Random Matrices from Traffic Probability
Limit Laws for Random Matrices from Traffic Probability arxiv:1601.02188 Slides available at math.berkeley.edu/ bensonau Benson Au UC Berkeley May 9th, 2016 Benson Au (UC Berkeley) Random Matrices from
More informationExpectation. DS GA 1002 Statistical and Mathematical Models. Carlos Fernandez-Granda
Expectation DS GA 1002 Statistical and Mathematical Models http://www.cims.nyu.edu/~cfgranda/pages/dsga1002_fall16 Carlos Fernandez-Granda Aim Describe random variables with a few numbers: mean, variance,
More informationBulk scaling limits, open questions
Bulk scaling limits, open questions Based on: Continuum limits of random matrices and the Brownian carousel B. Valkó, B. Virág. Inventiones (2009). Eigenvalue statistics for CMV matrices: from Poisson
More informationChapter 7 Polynomial Functions. Factoring Review. We will talk about 3 Types: ALWAYS FACTOR OUT FIRST! Ex 2: Factor x x + 64
Chapter 7 Polynomial Functions Factoring Review We will talk about 3 Types: 1. 2. 3. ALWAYS FACTOR OUT FIRST! Ex 1: Factor x 2 + 5x + 6 Ex 2: Factor x 2 + 16x + 64 Ex 3: Factor 4x 2 + 6x 18 Ex 4: Factor
More informationQuantum Probability and Asymptotic Spectral Analysis of Growing Roma, GraphsNovember 14, / 47
Quantum Probability and Asymptotic Spectral Analysis of Growing Graphs Nobuaki Obata GSIS, Tohoku University Roma, November 14, 2013 Quantum Probability and Asymptotic Spectral Analysis of Growing Roma,
More informationFinal Exam May 4, 2016
1 Math 425 / AMCS 525 Dr. DeTurck Final Exam May 4, 2016 You may use your book and notes on this exam. Show your work in the exam book. Work only the problems that correspond to the section that you prepared.
More informationFree Probability Theory and Random Matrices
Free Probability Theory and Random Matrices Roland Speicher Motivation of Freeness via Random Matrices In this chapter we want to motivate the definition of freeness on the basis of random matrices.. Asymptotic
More informationOperator-Valued Free Probability Theory and Block Random Matrices. Roland Speicher Queen s University Kingston
Operator-Valued Free Probability Theory and Block Random Matrices Roland Speicher Queen s University Kingston I. Operator-valued semicircular elements and block random matrices II. General operator-valued
More informationx 3y 2z = 6 1.2) 2x 4y 3z = 8 3x + 6y + 8z = 5 x + 3y 2z + 5t = 4 1.5) 2x + 8y z + 9t = 9 3x + 5y 12z + 17t = 7
Linear Algebra and its Applications-Lab 1 1) Use Gaussian elimination to solve the following systems x 1 + x 2 2x 3 + 4x 4 = 5 1.1) 2x 1 + 2x 2 3x 3 + x 4 = 3 3x 1 + 3x 2 4x 3 2x 4 = 1 x + y + 2z = 4 1.4)
More informationRecall that any inner product space V has an associated norm defined by
Hilbert Spaces Recall that any inner product space V has an associated norm defined by v = v v. Thus an inner product space can be viewed as a special kind of normed vector space. In particular every inner
More informationP AC COMMUTATORS AND THE R TRANSFORM
Communications on Stochastic Analysis Vol. 3, No. 1 (2009) 15-31 Serials Publications www.serialspublications.com P AC COMMUTATORS AND THE R TRANSFORM AUREL I. STAN Abstract. We develop an algorithmic
More informationMS 2001: Test 1 B Solutions
MS 2001: Test 1 B Solutions Name: Student Number: Answer all questions. Marks may be lost if necessary work is not clearly shown. Remarks by me in italics and would not be required in a test - J.P. Question
More informationExpectation. DS GA 1002 Probability and Statistics for Data Science. Carlos Fernandez-Granda
Expectation DS GA 1002 Probability and Statistics for Data Science http://www.cims.nyu.edu/~cfgranda/pages/dsga1002_fall17 Carlos Fernandez-Granda Aim Describe random variables with a few numbers: mean,
More informationTopics for the Qualifying Examination
Topics for the Qualifying Examination Quantum Mechanics I and II 1. Quantum kinematics and dynamics 1.1 Postulates of Quantum Mechanics. 1.2 Configuration space vs. Hilbert space, wave function vs. state
More informationSeparation of Variables in Linear PDE: One-Dimensional Problems
Separation of Variables in Linear PDE: One-Dimensional Problems Now we apply the theory of Hilbert spaces to linear differential equations with partial derivatives (PDE). We start with a particular example,
More informationDeterminantal point processes and random matrix theory in a nutshell
Determinantal point processes and random matrix theory in a nutshell part II Manuela Girotti based on M. Girotti s PhD thesis, A. Kuijlaars notes from Les Houches Winter School 202 and B. Eynard s notes
More informationA CHARACTERIZATION OF ULTRASPHERICAL POLYNOMIALS 1. THE QUESTION
A CHARACTERIZATION OF ULTRASPHERICAL POLYNOMIALS MICHAEL ANSHELEVICH ABSTRACT. We show that the only orthogonal polynomials with a generating function of the form F xz αz are the ultraspherical, Hermite,
More informationThe norm of polynomials in large random matrices
The norm of polynomials in large random matrices Camille Mâle, École Normale Supérieure de Lyon, Ph.D. Student under the direction of Alice Guionnet. with a significant contribution by Dimitri Shlyakhtenko.
More informationFFTs in Graphics and Vision. Fast Alignment of Spherical Functions
FFTs in Graphics and Vision Fast Alignment of Spherical Functions Outline Math Review Fast Rotational Alignment Review Recall 1: We can represent any rotation R in terms of the triplet of Euler angles
More informationDRAFT - Math 101 Lecture Note - Dr. Said Algarni
2 Limits 2.1 The Tangent Problems The word tangent is derived from the Latin word tangens, which means touching. A tangent line to a curve is a line that touches the curve and a secant line is a line that
More informationMOMENTS AND CUMULANTS OF THE SUBORDINATED LEVY PROCESSES
MOMENTS AND CUMULANTS OF THE SUBORDINATED LEVY PROCESSES PENKA MAYSTER ISET de Rades UNIVERSITY OF TUNIS 1 The formula of Faa di Bruno represents the n-th derivative of the composition of two functions
More informationChapter 1. Preliminaries. The purpose of this chapter is to provide some basic background information. Linear Space. Hilbert Space.
Chapter 1 Preliminaries The purpose of this chapter is to provide some basic background information. Linear Space Hilbert Space Basic Principles 1 2 Preliminaries Linear Space The notion of linear space
More informationPROBABILITY: LIMIT THEOREMS II, SPRING HOMEWORK PROBLEMS
PROBABILITY: LIMIT THEOREMS II, SPRING 218. HOMEWORK PROBLEMS PROF. YURI BAKHTIN Instructions. You are allowed to work on solutions in groups, but you are required to write up solutions on your own. Please
More informationCentral Limit Theorems for linear statistics for Biorthogonal Ensembles
Central Limit Theorems for linear statistics for Biorthogonal Ensembles Maurice Duits, Stockholm University Based on joint work with Jonathan Breuer (HUJI) Princeton, April 2, 2014 M. Duits (SU) CLT s
More informationApplications of the time derivative of the L 2 -Wasserstein distance and the free entropy dissipation
Applications of the time derivative of the L 2 -Wasserstein distance and the free entropy dissipation Hiroaki YOSHIDA Ochanomizu University Tokyo, Japan at Fields Institute 23 July, 2013 Plan of talk 1.
More informationLecture 4.6: Some special orthogonal functions
Lecture 4.6: Some special orthogonal functions Matthew Macauley Department of Mathematical Sciences Clemson University http://www.math.clemson.edu/~macaule/ Math 4340, Advanced Engineering Mathematics
More informationQualification Exam: Mathematical Methods
Qualification Exam: Mathematical Methods Name:, QEID#41534189: August, 218 Qualification Exam QEID#41534189 2 1 Mathematical Methods I Problem 1. ID:MM-1-2 Solve the differential equation dy + y = sin
More informationORTHOGONAL POLYNOMIALS IN PROBABILITY THEORY ABSTRACTS MINI-COURSES
ORTHOGONAL POLYNOMIALS IN PROBABILITY THEORY ABSTRACTS Jinho Baik (University of Michigan) ORTHOGONAL POLYNOMIAL ENSEMBLES MINI-COURSES TALK 1: ORTHOGONAL POLYNOMIAL ENSEMBLES We introduce a certain joint
More informationWorkshop on Free Probability and Random Combinatorial Structures. 6-8 December 2010
Workshop on Free Probability and Random Combinatorial Structures 6-8 December 2010 Bielefeld University Department of Mathematics V3-201 (Common Room) This workshop is part of the conference program of
More informationBoundary Conditions associated with the Left-Definite Theory for Differential Operators
Boundary Conditions associated with the Left-Definite Theory for Differential Operators Baylor University IWOTA August 15th, 2017 Overview of Left-Definite Theory A general framework for Left-Definite
More informationChapter 2 Polynomial and Rational Functions
Chapter 2 Polynomial and Rational Functions Section 1 Section 2 Section 3 Section 4 Section 5 Section 6 Section 7 Quadratic Functions Polynomial Functions of Higher Degree Real Zeros of Polynomial Functions
More informationProbability and combinatorics
Texas A&M University May 1, 2012 Probability spaces. (Λ, M, P ) = measure space. Probability space: P a probability measure, P (Λ) = 1. Probability spaces. (Λ, M, P ) = measure space. Probability space:
More informationBrownian Motion and Stochastic Calculus
ETHZ, Spring 17 D-MATH Prof Dr Martin Larsson Coordinator A Sepúlveda Brownian Motion and Stochastic Calculus Exercise sheet 6 Please hand in your solutions during exercise class or in your assistant s
More informationProgress in the method of Ghosts and Shadows for Beta Ensembles
Progress in the method of Ghosts and Shadows for Beta Ensembles Alan Edelman (MIT) Alex Dubbs (MIT) and Plamen Koev (SJS) Oct 8, 2012 1/47 Wishart Matrices (arbitrary covariance) G=mxn matrix of Gaussians
More informationFree Entropy for Free Gibbs Laws Given by Convex Potentials
Free Entropy for Free Gibbs Laws Given by Convex Potentials David A. Jekel University of California, Los Angeles Young Mathematicians in C -algebras, August 2018 David A. Jekel (UCLA) Free Entropy YMC
More informationADVANCED ENGINEERING MATHEMATICS MATLAB
ADVANCED ENGINEERING MATHEMATICS WITH MATLAB THIRD EDITION Dean G. Duffy Contents Dedication Contents Acknowledgments Author Introduction List of Definitions Chapter 1: Complex Variables 1.1 Complex Numbers
More information256 Summary. D n f(x j ) = f j+n f j n 2n x. j n=1. α m n = 2( 1) n (m!) 2 (m n)!(m + n)!. PPW = 2π k x 2 N + 1. i=0?d i,j. N/2} N + 1-dim.
56 Summary High order FD Finite-order finite differences: Points per Wavelength: Number of passes: D n f(x j ) = f j+n f j n n x df xj = m α m dx n D n f j j n= α m n = ( ) n (m!) (m n)!(m + n)!. PPW =
More informationELEMENTS OF PROBABILITY THEORY
ELEMENTS OF PROBABILITY THEORY Elements of Probability Theory A collection of subsets of a set Ω is called a σ algebra if it contains Ω and is closed under the operations of taking complements and countable
More informationMFM Practitioner Module: Quantitative Risk Management. John Dodson. September 23, 2015
MFM Practitioner Module: Quantitative Risk Management September 23, 2015 Mixtures Mixtures Mixtures Definitions For our purposes, A random variable is a quantity whose value is not known to us right now
More informationPartial Differential Equations
Part II Partial Differential Equations Year 2015 2014 2013 2012 2011 2010 2009 2008 2007 2006 2005 2015 Paper 4, Section II 29E Partial Differential Equations 72 (a) Show that the Cauchy problem for u(x,
More informationMathematics (MA) Mathematics (MA) 1. MA INTRO TO REAL ANALYSIS Semester Hours: 3
Mathematics (MA) 1 Mathematics (MA) MA 502 - INTRO TO REAL ANALYSIS Individualized special projects in mathematics and its applications for inquisitive and wellprepared senior level undergraduate students.
More information5.1 Consistency of least squares estimates. We begin with a few consistency results that stand on their own and do not depend on normality.
88 Chapter 5 Distribution Theory In this chapter, we summarize the distributions related to the normal distribution that occur in linear models. Before turning to this general problem that assumes normal
More informationSpectral Theory of X 1 -Laguerre Polynomials
Advances in Dynamical Systems and Applications ISSN 973-5321, Volume 8, Number 2, pp. 181 192 (213) http://campus.mst.edu/adsa Spectral Theory of X 1 -Laguerre Polynomials Mohamed J. Atia Université de
More informationClassical Fourier Analysis
Loukas Grafakos Classical Fourier Analysis Second Edition 4y Springer 1 IP Spaces and Interpolation 1 1.1 V and Weak IP 1 1.1.1 The Distribution Function 2 1.1.2 Convergence in Measure 5 1.1.3 A First
More informationMath 651 Introduction to Numerical Analysis I Fall SOLUTIONS: Homework Set 1
ath 651 Introduction to Numerical Analysis I Fall 2010 SOLUTIONS: Homework Set 1 1. Consider the polynomial f(x) = x 2 x 2. (a) Find P 1 (x), P 2 (x) and P 3 (x) for f(x) about x 0 = 0. What is the relation
More informationMA/ST 810 Mathematical-Statistical Modeling and Analysis of Complex Systems
MA/ST 810 Mathematical-Statistical Modeling and Analysis of Complex Systems Review of Basic Probability The fundamentals, random variables, probability distributions Probability mass/density functions
More informationVectors in Function Spaces
Jim Lambers MAT 66 Spring Semester 15-16 Lecture 18 Notes These notes correspond to Section 6.3 in the text. Vectors in Function Spaces We begin with some necessary terminology. A vector space V, also
More informationApplied Linear Algebra
Applied Linear Algebra Peter J. Olver School of Mathematics University of Minnesota Minneapolis, MN 55455 olver@math.umn.edu http://www.math.umn.edu/ olver Chehrzad Shakiban Department of Mathematics University
More informationn E(X t T n = lim X s Tn = X s
Stochastic Calculus Example sheet - Lent 15 Michael Tehranchi Problem 1. Let X be a local martingale. Prove that X is a uniformly integrable martingale if and only X is of class D. Solution 1. If If direction:
More informationThe circular law. Lewis Memorial Lecture / DIMACS minicourse March 19, Terence Tao (UCLA)
The circular law Lewis Memorial Lecture / DIMACS minicourse March 19, 2008 Terence Tao (UCLA) 1 Eigenvalue distributions Let M = (a ij ) 1 i n;1 j n be a square matrix. Then one has n (generalised) eigenvalues
More informationProbability and Estimation. Alan Moses
Probability and Estimation Alan Moses Random variables and probability A random variable is like a variable in algebra (e.g., y=e x ), but where at least part of the variability is taken to be stochastic.
More informationLINEAR ALGEBRA BOOT CAMP WEEK 4: THE SPECTRAL THEOREM
LINEAR ALGEBRA BOOT CAMP WEEK 4: THE SPECTRAL THEOREM Unless otherwise stated, all vector spaces in this worksheet are finite dimensional and the scalar field F is R or C. Definition 1. A linear operator
More informationLecture 22: Variance and Covariance
EE5110 : Probability Foundations for Electrical Engineers July-November 2015 Lecture 22: Variance and Covariance Lecturer: Dr. Krishna Jagannathan Scribes: R.Ravi Kiran In this lecture we will introduce
More informationCombinatorial Aspects of Free Probability and Free Stochastic Calculus
Combinatorial Aspects of Free Probability and Free Stochastic Calculus Roland Speicher Saarland University Saarbrücken, Germany supported by ERC Advanced Grant Non-Commutative Distributions in Free Probability
More informationEXACT SOLUTIONS for ORDINARY DIFFERENTIAL EQUATIONS SECOND EDITION
HANDBOOK OF EXACT SOLUTIONS for ORDINARY DIFFERENTIAL EQUATIONS SECOND EDITION Andrei D. Polyanin Valentin F. Zaitsev CHAPMAN & HALL/CRC A CRC Press Company Boca Raton London New York Washington, D.C.
More informationRandom Variables. P(x) = P[X(e)] = P(e). (1)
Random Variables Random variable (discrete or continuous) is used to derive the output statistical properties of a system whose input is a random variable or random in nature. Definition Consider an experiment
More informationTriangular matrices and biorthogonal ensembles
/26 Triangular matrices and biorthogonal ensembles Dimitris Cheliotis Department of Mathematics University of Athens UK Easter Probability Meeting April 8, 206 2/26 Special densities on R n Example. n
More informationMATHEMATICS. Course Syllabus. Section A: Linear Algebra. Subject Code: MA. Course Structure. Ordinary Differential Equations
MATHEMATICS Subject Code: MA Course Structure Sections/Units Section A Section B Section C Linear Algebra Complex Analysis Real Analysis Topics Section D Section E Section F Section G Section H Section
More informationContents. 1 Preliminaries 3. Martingales
Table of Preface PART I THE FUNDAMENTAL PRINCIPLES page xv 1 Preliminaries 3 2 Martingales 9 2.1 Martingales and examples 9 2.2 Stopping times 12 2.3 The maximum inequality 13 2.4 Doob s inequality 14
More informationCumulants of a convolution and applications to monotone probability theory
Cumulants of a convolution and applications to monotone probability theory arxiv:95.3446v2 [math.pr] 24 May 29 Takahiro Hasebe JSPS Research Fellow (DC), Graduate School of Science, Kyoto University, Kyoto
More information18.440: Lecture 28 Lectures Review
18.440: Lecture 28 Lectures 18-27 Review Scott Sheffield MIT Outline Outline It s the coins, stupid Much of what we have done in this course can be motivated by the i.i.d. sequence X i where each X i is
More informationMathematics of Physics and Engineering II: Homework answers You are encouraged to disagree with everything that follows. P R and i j k 2 1 1
Mathematics of Physics and Engineering II: Homework answers You are encouraged to disagree with everything that follows Homework () P Q = OQ OP =,,,, =,,, P R =,,, P S = a,, a () The vertex of the angle
More information9th Sendai Workshop Infinite Dimensional Analysis and Quantum Probability
9th Sendai Workshop Infinite Dimensional Analysis and Quantum Probability September 11-12, 2009 Graduate School of Information Sciences Tohoku University PROGRAM September 11 (Fri) GSIS, Large Lecture
More information18.175: Lecture 17 Poisson random variables
18.175: Lecture 17 Poisson random variables Scott Sheffield MIT 1 Outline More on random walks and local CLT Poisson random variable convergence Extend CLT idea to stable random variables 2 Outline More
More informationMcGill University Department of Mathematics and Statistics. Ph.D. preliminary examination, PART A. PURE AND APPLIED MATHEMATICS Paper BETA
McGill University Department of Mathematics and Statistics Ph.D. preliminary examination, PART A PURE AND APPLIED MATHEMATICS Paper BETA 17 August, 2018 1:00 p.m. - 5:00 p.m. INSTRUCTIONS: (i) This paper
More informationASSIGNMENT-1 M.Sc. (Previous) DEGREE EXAMINATION, DEC First Year MATHEMATICS. Algebra. MAXIMUM MARKS:30 Answer ALL Questions
(DM1) ASSIGNMENT-1 Algebra MAXIMUM MARKS:3 Q1) a) If G is an abelian group of order o(g) and p is a prime number such that p α / o(g), p α+ 1 / o(g) then prove that G has a subgroup of order p α. b) State
More informationVector Spaces. Vector space, ν, over the field of complex numbers, C, is a set of elements a, b,..., satisfying the following axioms.
Vector Spaces Vector space, ν, over the field of complex numbers, C, is a set of elements a, b,..., satisfying the following axioms. For each two vectors a, b ν there exists a summation procedure: a +
More informationOctavio Arizmendi, Ole E. Barndorff-Nielsen and Víctor Pérez-Abreu
ON FREE AND CLASSICAL TYPE G DISTRIBUTIONS Octavio Arizmendi, Ole E. Barndorff-Nielsen and Víctor Pérez-Abreu Comunicación del CIMAT No I-9-4/3-4-29 (PE /CIMAT) On Free and Classical Type G Distributions
More informationOrthogonal Polynomial Ensembles
Chater 11 Orthogonal Polynomial Ensembles 11.1 Orthogonal Polynomials of Scalar rgument Let wx) be a weight function on a real interval, or the unit circle, or generally on some curve in the comlex lane.
More informationLyapunov exponents of free operators
Journal of Functional Analysis 255 (28) 1874 1888 www.elsevier.com/locate/jfa Lyapunov exponents of free operators Vladislav Kargin Department of Mathematics, Stanford University, Stanford, CA 9435, USA
More informationMathematical Methods for Engineers and Scientists 1
K.T. Tang Mathematical Methods for Engineers and Scientists 1 Complex Analysis, Determinants and Matrices With 49 Figures and 2 Tables fyj Springer Part I Complex Analysis 1 Complex Numbers 3 1.1 Our Number
More informationThe concentration of a drug in blood. Exponential decay. Different realizations. Exponential decay with noise. dc(t) dt.
The concentration of a drug in blood Exponential decay C12 concentration 2 4 6 8 1 C12 concentration 2 4 6 8 1 dc(t) dt = µc(t) C(t) = C()e µt 2 4 6 8 1 12 time in minutes 2 4 6 8 1 12 time in minutes
More informationMultiple orthogonal polynomials. Bessel weights
for modified Bessel weights KU Leuven, Belgium Madison WI, December 7, 2013 Classical orthogonal polynomials The (very) classical orthogonal polynomials are those of Jacobi, Laguerre and Hermite. Classical
More informationLuigi Accardi Classification of probability measures in terms of the canonically associated commutation relations Talk given at the: 5-th Levy
Luigi Accardi Classification of probability measures in terms of the canonically associated commutation relations Talk given at the: 5-th Levy Conference (for the 20-th centenary of P. Levy) Meijo University,
More information