期刊名称：Latin American Journal of Probability and Mathematical Statistics
电子版ISSN：1980-0436
出版年度：2012
卷号：IX
期号：2
页码：685-715
出版社：Instituto Nacional De Matemática Pura E Aplicada
摘要：The celebrated Marˇcenko-Pastur theorem gives the asymptotic spectral distribution ofsums of random, independent, rank-one projections. Its main hypothesis is that these projectionsare more or less uniformly distributed on the first grassmannian, which implies for example thatthe corresponding vectors are delocalized, i.e. are essentially supported by the whole canonicalbasis. In this paper, we propose a way to drop this delocalization assumption and we generalize thistheorem to a quite general framework, including random projections whose corresponding vectorsare localized, i.e. with some components much larger than the other ones. The first of our twomain examples is given by heavy tailed random vectors (as in the model introduced by Ben Arousand Guionnet (2008) or as in the model introduced by Zakharevich (2006) where the momentsgrow very fast as the dimension grows). Our second main example, related to the continuumbetween the classical and free convolutions introduced in Benaych-Georges and L´evy (2011), isgiven by vectors which are distributed as the Brownian motion on the unit sphere, with localizedinitial law. Our framework is in fact general enough to get new correspondences between classicalinfinitely divisible laws and some limit spectral distributions of random matrices, generalizing theso-called Bercovici-Pata bijection.
