  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.
