Marchenko-Pastur Law for Tyler’s and Maronna’s M-estimators

Marčenko-Pastur Law for Tyler’s and Maronna’s M-estimators

This paper studies the limiting behavior of Tyler’s and Maronna’s Mestimators, in the regime that the number of samples n and the dimension p both go to infinity, and p/n converges to a constant y with 0 < y < 1. We prove that when the data samples are identically and independently generated from the Gaussian distribution N(0, I), the difference between 1 n ∑n i=1 xix T i and a scaled version o...

On the Convergence of Maronna's M-Estimators of Scatter

In this paper, we propose an alternative proof for the uniqueness of Maronna’s M -estimator of scatter [1] for N vector observations y1, . . . ,yN ∈ R under a mild constraint of linear independence of any subset of m of these vectors. This entails in particular almost sure uniqueness for random vectors yi with a density as long as N > m. This approach allows to establish further relations that ...

On a surprising relation between the Marchenko-Pastur law, rectangular and square free convolutions

In this paper, we prove a result linking the square and the rectangular R-transforms, which consequence is a surprising relation between the square and rectangular free convolutions, involving the Marchenko-Pastur law. Consequences on infinite divisibility and on the arithmetics of Voiculescu’s free additive and multiplicative convolutions are given.

A total positivity property of the Marchenko-Pastur Law

M ar 2 00 7 RANDOM MATRICES , NON - BACKTRACKING WALKS , AND ORTHOGONAL POLYNOMIALS

Several well-known results from the random matrix theory, such as Wigner’s law and the Marchenko–Pastur law, can be interpreted (and proved) in terms of non-backtracking walks on a certain graph. Orthogonal polynomials with respect to the limiting spectral measure play a rôle in this approach.