Noureddine El Karoui

Tracy–Widom limit for the largest eigenvalue of a large class of complex sample covariance matrices

We consider the asymptotic fluctuation behavior of the largest eigenvalue of certain sample covariance matrices in the asymptotic regime where both dimensions of the corresponding data matrix go to infinity. More precisely, let X be an n x p matrix, and let its rows be i.i.d. complex normal vectors with mean 0 and covariance Σ p . We show that for a large class of covariance matrices £ p, the largest eigenvalue of X*X is asymptotically distributed (after recentering and rescaling) as the Tracy-Widom distribution that appears in the study of the Gaussian unitary ensemble. We give explicit formulas for the centering and scaling sequences that are easy to implement and involve only the spectral distribution of the population covariance, n and p. The main theorem applies to a number of covariance models found in applications. For example, well-behaved Toeplitz matrices as well as covariance matrices whose spectral distribution is a sum of atoms (under some conditions on the mass of the atoms) are among the models the theorem can handle. Generalizations of the theorem to certain spiked versions of our models and a.s. results about the largest eigenvalue are given. We also discuss a simple corollary that does not require normality of the entries of the data matrix and some consequences for applications in multivariate statistics.

A rate of convergence result for the largest eigenvalue of complex white Wishart matrices

It has been recently shown that if X is an n×N matrix whose entries are i.i.d. standard complex Gaussian and l1 is the largest eigenvalue of X*X, there exist sequences mn,N and sn,N such that (l1−mn,N)/sn,N converges in distribution to W2, the Tracy–Widom law appearing in the study of the Gaussian unitary ensemble. This probability law has a density which is known and computable. The cumulative distribution function of W2 is denoted F2. In this paper we show that, under the assumption that n/N→ γ∈(0, ∞), we can find a function M, continuous and nonincreasing, and sequences μn,N and σn,N such that, for all real s0, there exists an integer N(s0, γ) for which, if (n∧N)≥N(s0, γ), we have, with ln,N=(l1−μn,N)/σn,N, ∀ s≥s0 (n∧N)2/3|P(ln,N≤s)−F2(s)|≤M(s0)exp(−s). The surprisingly good 2/3 rate and qualitative properties of the bounding function help explain the fact that the limiting distribution W2 is a good approximation to the empirical distribution of ln,N in simulations, an important fact from the point of view of (e.g., statistical) applications.

Concentration of measure and spectra of random matrices: Applications to correlation matrices, elliptical distributions and beyond.

We place ourselves in the setting of high-dimensional statistical inference, where the number of variables

On the Realized Risk of High-Dimensional Markowitz Portfolios

p

On robust regression with high-dimensional predictors

in a data set of interest is of the same order of magnitude as the number of observations

Optimal M-estimation in high-dimensional regression

n

Optimizing Automated Classification of Variable Stars in New Synoptic Surveys

. More formally, we study the asymptotic properties of correlation and covariance matrices, in the setting where

Machine Learning and Portfolio Optimization

p/n\to\rho\in(0,\infty),

Second order accurate distributed eigenvector computation for extremely large matrices

for general population covariance. We show that, for a large class of models studied in random matrix theory, spectral properties of large-dimensional correlation matrices are similar to those of large-dimensional covarance matrices. We also derive a Mar\u{c}enko--Pastur-type system of equations for the limiting spectral distribution of covariance matrices computed from data with elliptical distributions and generalizations of this family. The motivation for this study comes partly from the possible relevance of such distributional assumptions to problems in econometrics and portfolio optimization, as well as robustness questions for certain classical random matrix results. A mathematical theme of the paper is the important use we make of concentration inequalities.

Weak Recovery Conditions from Graph Partitioning Bounds and Order Statistics

We study the realized risk of Markowitz portfolios computed using parameters estimated from data and generalizations to similar questions involving the out-of-sample risk in quadratic programs with linear equality constraints. We do so under the assumption that the data is generated according to an elliptical model, which allows us to study models with heavy tails, tail dependence, and leptokurtic marginals for the data. We place ourselves in the setting of high-dimensional inference where the number of assets in the portfolio,