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21
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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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.
STEIN’S METHOD AND CHARACTERS OF COMPACT LIE GROUPS
, 2008
"... Stein’s method is used to study the trace of a random element from a compact Lie group or symmetric space. Central limit theorems are proved using very little information: character values on a single element and the decomposition of the square of the trace into irreducible components. This is illu ..."
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Stein’s method is used to study the trace of a random element from a compact Lie group or symmetric space. Central limit theorems are proved using very little information: character values on a single element and the decomposition of the square of the trace into irreducible components. This is illustrated for Lie groups of classical type and Dyson’s circular ensembles. The approach in this paper will be useful for the study of higher dimensional characters, where normal approximations need not hold.
Semicircle law for a class of random matrices with dependent entries. arXiv preprint arXiv:1211.0389
, 2012
"... Abstract. In this paper we study ensembles of random symmetric matrices Xn = {Xij}ni,j=1 with a random field type dependence, such that EXij = 0, EX2ij = σ2ij, where σij can be different numbers. Assuming that the average of the normalized sums of variances in each row converges to one and Lindeberg ..."
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Abstract. In this paper we study ensembles of random symmetric matrices Xn = {Xij}ni,j=1 with a random field type dependence, such that EXij = 0, EX2ij = σ2ij, where σij can be different numbers. Assuming that the average of the normalized sums of variances in each row converges to one and Lindeberg condition holds true we prove that the empirical spectral distribution of eigenvalues converges to Wigner’s semicircle law.
Vector diffusion maps and random matrices with random blocks
, 2014
"... Vector diffusion maps (VDM) is a modern data analysis technique that is starting to be applied for the analysis of high dimensional and massive datasets. Motivated by this technique, we study matrices that are akin to the ones appearing in the null case of VDM, i.e the case where there is no structu ..."
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Vector diffusion maps (VDM) is a modern data analysis technique that is starting to be applied for the analysis of high dimensional and massive datasets. Motivated by this technique, we study matrices that are akin to the ones appearing in the null case of VDM, i.e the case where there is no structure in the dataset under investigation. Developing this understanding is important in making sense of the output of the VDM algorithm whether there is signal or not. We hence develop a theory explaining the behavior of the spectral distribution of a large class of random matrices, in particular random matrices with random block entries. Numerical work shows that the agreement between our theoretical predictions and numerical simulations is generally very good. 1
STEIN’S METHOD, SEMICIRCLE DISTRIBUTION, AND REDUCED DECOMPOSITIONS OF THE LONGEST ELEMENT IN THE SYMMETRIC GROUP
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Limiting spectral distribution for wigner matrices with dependent entries. arXiv preprint arXiv:1304.3394
, 2013
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ON THE RATE OF CONVERGENCE AND BERRYESSEEN TYPE THEOREMS FOR A MULTIVARIATE FREE CENTRAL LIMIT THEOREM
"... Abstract. We address the question of a Berry Esseen type theorem for the speed of convergence in a multivariate free central limit theorem. For this, we estimate the difference between the operatorvalued Cauchy transforms of the normalized partial sums in an operatorvalued free central limit theor ..."
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Abstract. We address the question of a Berry Esseen type theorem for the speed of convergence in a multivariate free central limit theorem. For this, we estimate the difference between the operatorvalued Cauchy transforms of the normalized partial sums in an operatorvalued free central limit theorem and the Cauchy transform of the limiting operatorvalued semicircular element. 1.