Marchenko-Pastur Theorem and Bercovici-Pata bijections for heavy-tailed or localized vectors

TL;DR

This paper extends the Marchenko-Pastur theorem to include heavy-tailed and localized vectors, establishing new links between classical infinitely divisible laws and spectral distributions of random matrices.

Contribution

It generalizes the Marchenko-Pastur theorem by removing the delocalization assumption, applying to heavy-tailed and localized vectors, and extends Bercovici-Pata bijections to this broader setting.

Findings

Generalized spectral distribution results for heavy-tailed vectors

Established new Bercovici-Pata type correspondences

Included models with localized vectors and Brownian motion on spheres

Abstract

The celebrated Marchenko-Pastur theorem gives the asymptotic spectral distribution of sums of random, independent, rank-one projections. Its main hypothesis is that these projections are more or less uniformly distributed on the first grassmannian, which implies for example that the corresponding vectors are delocalized, i.e. are essentially supported by the whole canonical basis. In this paper, we propose a way to drop this delocalization assumption and we generalize this theorem to a quite general framework, including random projections whose corresponding vectors are localized, i.e. with some components much larger than the other ones. The first of our two main examples is given by heavy tailed random vectors (as in a model introduced by Ben Arous and Guionnet or as in a model introduced by Zakharevich where the moments grow very fast as the dimension grows). Our second main example is given by vectors which are distributed as the Brownian motion on the unit sphere, with localized initial law. Our framework is in fact general enough to get new correspondences between classical infinitely divisible laws and some limit spectral distributions of random matrices, generalizing the so-called Bercovici-Pata bijection.