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A law of large numbers for finiterange dependent random matrices
 Comm. Pure Appl. Math
"... Abstract. We consider random hermitian matrices in which distant abovediagonal entries are independent but nearby entries may be correlated. We find the limit of the empirical distribution of eigenvalues by combinatorial methods. We also prove that the limit has algebraic Stieltjes transform by an a ..."
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Abstract. We consider random hermitian matrices in which distant abovediagonal entries are independent but nearby entries may be correlated. We find the limit of the empirical distribution of eigenvalues by combinatorial methods. We also prove that the limit has algebraic Stieltjes transform by an argument based on dimension theory of noetherian local rings. 1.
2012): MarchenkoPastur theorem and BercoviciPata bijections for heavytailed or localized vectors
 ALEA, Latin American J. Probab. Statist
"... Abstract. The celebrated MarchenkoPasturtheorem gives the asymptotic spectral distribution of sums of random, independent, rankone projections. Its main hypothesis is that these projections are more or less uniformly distributed on the first grassmannian, which implies for example that the corresp ..."
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Cited by 2 (1 self)
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Abstract. The celebrated MarchenkoPasturtheorem gives the asymptotic spectral distribution of sums of random, independent, rankone 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 wayto 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 byheavytailed randomvectors(asin the modelintroduced byBen Arousand Guionnet in [5] or as in the model introduced by Zakharevichin [32] where the moments grow very fast as the dimension grows). Our second main example, related to the continuum between the classical and free convolutions introduced in [11], 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 socalled BercoviciPata bijection. 1.
MarčenkoPastur theorem and BercoviciPata bijections for heavytailed or localized vectors
, 2012
"... Abstract. The celebrated MarčenkoPastur theorem gives the asymptotic spectral distribution of sums of random, independent, rankone projections. Its main hypothesis is that these projections are more or less uniformly distributed on the first grassmannian, which implies for example that the corresp ..."
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Abstract. The celebrated MarčenkoPastur theorem gives the asymptotic spectral distribution of sums of random, independent, rankone 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 the model introduced by Ben Arous and Guionnet (2008) or as in the model introduced by Zakharevich (2006) where the moments grow very fast as the dimension grows). Our second main example, related to the continuum between the classical and free convolutions introduced in BenaychGeorges and Lévy (2011), 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 socalled BercoviciPata bijection. 1.