Random Matrix Theory and its Applications to Statistics and Wireless Communications
(26 Feb - 31 Mar 2006)
~ Abstracts ~
Eigenvalues of large sample
covariance matrices
Jinho Baik, University of Michigan, USA
Suppose that the dimension p of each sample (population size) is large and is comparable to the size of the number n of samples. Functional data such as voice or image samples is a such example. In this case, even for the independent Gaussian samples, the eigenvalues of the sample covariance matrix are not close to 1, but they spread out in an interval that can be determined by the ratio p/n. Moreover with probability 1, all the sample eigenvalues lie inside the interval. What happens if the true covariance matrix is not identity, but rather a finite-rank perturbation of the identity matrix (hence only finitely many non-unit true eigenvalues)? A recent study shows that there is a critical strength of the non-unit eigenvalues that distort the interval structure of the eigenvalues. Hence even when both p and n are large, one may detect large true eigenvalues from the sample eigenvalues under the assumption we made if it is larger than a critical value.In this series of lectures, we will discuss on this question for the complex Gaussian case. It will be demonstrated that for complex Gaussian samples, one can analyze the asymptotics of the eigenvalues in detail so that one can obtain the limit laws of the largest eigenvalues for all possible values of the non-unit true eigenvalues. The lecture will try to be self-contained and elementary for graduate students.
Lecture 1.
Basics Sample covariance matrix, Eigenvalue density
function, Correlation functions, Distribution function for
the largest eigenvalue, Meaning of asymptotics: large
population size, large sample size. Results for the sample
eigenvalues for independent Gaussian samples, Spiked
population model where the true covariance matrix is a
finite-rank perturbation of the identity matrix
Lecture 2.
Asymptotic analysis Steepest-descent method Airy operator, A
generalization of the Airy operator, Transition
Lecture 3.
Differential equations A search for Tracy-Widom type formula
for the generalization of the Airy operator
Lecture 4.
Other models What if the true covariance matrix has two
eigenvalues of high multiplicity? Interacting particle
systems (traffic model) Queues in tandem
A model for the bus system in
Cuernevaca (Mexico)
Jinho Baik, University of Michigan, USA
Abstract: The bus system in Cuernevanca, Mexico and its connections to random matrix distributions has been the subject of an interesting recent study by Krbalek and Seba. We introduce and analyze a microscopic model. This is a joint work with Alexei Borodin, Percy Deift and Toufic Suidan.
The zeros of the Riemann zeta
function and random matrix theory
Zeev Rudnick, Tel-Aviv University, Israel
The zeros of the Riemann zeta function are amongst the most fundamental entities in number theory. Relatively recently, it has been discovered that their statistics are the same as those of certain random matrix ensemble - the CUE. In the talk I will explain the relevance of studying zeros of Riemann's zeta function to understanding primes, what is the Riemann Hypothesis, and what is known concerning the statistics of the zeros of the zeros and the connection with Random Matrix Theory. This will be accompanied by an exposition of the large body of numerical evidence collected to date. I will assume no prior knowledge of number theory and the Riemann zeta function.
On limit theorem for the
Eigenvalues of product of two random matrices
Baisuo Jin, University of Science and Technology of China
In the paper, we give a strict proof for existence of the limiting spectral distribution of a product of a sample covariance with a Hermitian random matrix. It is well known that the limiting spectral distribution of such a product exists when the Hermitian matrix is p.d. which is crucial in previous works. We proved this without this crucial restriction. Especially, we derived the explicit form of the limiting spectral distribution when the Hermitian matrix is a Wigner matrix which becomes the 8th product beyond 7 with known explicit forms of densities of their limiting spectral distributions.
Making Markowitz’s portfolio
principle practically useful
Huixia Liu, National University of Singapore
The Markowitz mean-variance optimization procedure to compute the optimal return is highly appreciated as a one of the most important cornerstones in modern finance theory. However, the traditional estimated return has been demonstrated not to be applicable in practice due to its serious departure from its theoretic optimal return, attributed to the substantial measurement error. Applying the theory of large dimensional data analysis, we first theoretically explain this phenomenon is natural when the number of assets is large. We also show that the huge measurement error is due to the serious departure of the estimated asset allocation from its theoretic counterpart. Thereafter, we prove that the estimated optimal return is always larger than its theoretic parameter when the number of assets is large. To circumvent this problem, we utilize both the large dimensional random matrix theory and the parametric bootstrap method to develop new bootstrap estimators for the optimal return and its asset allocation. We further theoretically prove that these bootstrap estimates are consistent to their counterpart parameters. Our simulation confirms the consistency and shows that, comparing with the traditional estimate, our proposed estimate improves the estimation accuracy so substantial that its relative efficiency is as high as 139 times for sample size of 500; implying that the essence of the portfolio analysis problem could be adequately captured by our proposed estimates. The improvement of our proposed estimates are so big that there is a sound basis for believing our proposed estimates to be the best estimates to date that they greatly enhance the Markowitz mean-variance optimization procedure to be implementable and practically useful.
Spectral measure of large Hankel,
Markov and Toeplitz matrices
Tiefeng Jiang, University of Minnesota
We study the limiting spectral measure of large symmetric random matrices. This includes the asymptotic behavior of properly scaled eigenvalues of Hankel, Markov and Toeplitz matrices. This solves three unsolved random matrix problems. It is the joint work with W. Bryc and A. Dembo.
How many entries of a typical
orthogonal matrix can be approximated by independent
normals?
Tiefeng Jiang, University of Minnesota
I will present my solution to the open problem by Diaconis: what are the largest orders of the upper left block of a random matrix which is uniformly distributed on the orthogonal groups, can be approximated by independent standard normals? This problem is solved by two different approximation methods: the variation norm and a weak norm. The history of the problem since 1906 will be reviewed; connections to Engineering, Mechanics, Statistics, Probability and Mathematics will be presented; applications and future problems will also be given in this talk.
A new method to bound rate of
convergence
Arup Bose, Indian Statistical Institute
When the Empirical Spectral Distribution converges, one may like to know the rate of convergence. In this talk we explain a general technique of bounding the rate of convergence to the Limiting Spectral Distribution. We show how our results apply to the Wigner matrix and to the Sample Covariance matrix.
Another look at the moment method
and some new results
Arup Bose, Indian Statistical Institute
We first quickly review the method of Bryc, Jiang and Dembo which has been used to obtain the Limiting Spectral Distribution of the Toeplitz and related matrices. Then we show how this method leads to a "unified" approach to establish Limiting Spectral Distributions. We also show how it can be adapted to obtaining some new results for matrices with dependent entries.
Eigenvalue distribution of a
class of Gram random matrices and applications in wireless
communications
Walid Hachem, Ecole Supérieure d'Electricité, France
Consider a N x n random matrix Q = Y + A where Y is a random matrix with centered independent elements having a variance profile and A is a deterministic matrix. We begin by studying the eigenvalue distribution of the Gram matrix Q Q'. Except in some special cases, this distribution does not converge when N grows toward infinity and N / n converges to a positive constant. Following an idea of Girko, it is however possible to obtain a deterministic approximation of the eigenvalue distribution for finite values of N. This work is motivated by the problem of Shannon's mutual information evaluation for Multiple Input Multiple Output wireless communication channels. As an application, we derive a deterministic approximation of the mutual information (1/N) log det ( 1 + Q Q'/ sigma^2 ) where sigma^2 is a known parameter. Some results concerning a Central Limit Theorem for the mutual information are also given.
Selberg's multidimensional beta
integral and some extensions of it
Richard Askey, University of Wisconsin-Madison
Selberg's beta integral sat in obscurity for a few decades. Only after a limiting case of it arose in work of Dyson and Mehta did many people become interested in such integrals. There are now too many to talk on in one hour, so a sketch of a proof of Aomoto's extension will be given and some other cases will be summarized.
Some examples of results on
orthogonal polynomials
Richard Askey, University of Wisconsin-Madison
Depending on the interest of the audience some useful results on orthogonal polynomials will be described. One is an extension of a result of Sylvester on the eigenvalues of the tri-diagonal matrix with zeros on the main diagonal, 1,2,...,N above this diagonal and N,N-1,...,1 below this diagonal. Others will be decided after I have had a chance to talk with some at this meeting.
Singular linear statistics in GUE
and introduction to the Tracy-Widom distribution
Yang Chen, Nankai University, China and Imperial College
London, UK
This talk will focus on the computation of a certain discontinuous linear statistics in the Gaussian Unitary Ensemble (GUE), which give rise to orthogonal polynomials with discontinuous weights, and leads to a particular Painleve IV for the distribution function. The Tracy-Widom or the largest eigenvalue distribution of the GUE is re-derived here as an application of the ladder operator method to be described in the talk.
Beta generalizations of the
classical random matrix ensembles
Peter Forrester, University of Melbourne, Australia
Lecture 1: Classical random matrix ensembles from physical applications
Lecture 2: Calculation of the eigenvalue PDF for the classical ensembles
Lecture 3: The Gaussian beta ensemble
Lecture 4: The Laguerre beta ensemble
Partition function of one-matrix
models in the multi-cut case
Tamara Grava, SISSA-ISAS, Italy
In this talk we will present some numerical simulations of the partition function of one matrix models in the large N limit when the corresponding eigenvalues are distributed on many disjoint intervals. We compare these results with the numerical simulations obtained for the small dispersion limit of the Korteweg de Vries equation.
