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47
Integration over the Pauli quantum group
"... Abstract. We prove that the Pauli representation of the quantum permutation algebra A(S4) is faithful. This provides the second known model for a free quantum algebra. We use this model for performing some computations, with the result that at level of laws of diagonal coordinates, the Lebesgue meas ..."
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Abstract. We prove that the Pauli representation of the quantum permutation algebra A(S4) is faithful. This provides the second known model for a free quantum algebra. We use this model for performing some computations, with the result that at level of laws of diagonal coordinates, the Lebesgue measure appears between the Dirac mass and the free Poisson law.
Concentration of measure and spectra of random matrices: with applications to correlation matrices, elliptical distributions and beyond
 THE ANNALS OF APPLIED PROBABILITY TO APPEAR
, 2009
"... We place ourselves in the setting of highdimensional statistical inference, where the number of variables p in a dataset of interest is of the same order of magnitude as the number of observations n. More formally we study the asymptotic properties of correlation and covariance matrices under the s ..."
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Cited by 8 (5 self)
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We place ourselves in the setting of highdimensional statistical inference, where the number of variables p in a dataset of interest is of the same order of magnitude as the number of observations n. More formally we study the asymptotic properties of correlation and covariance matrices under the setting that p/n → ρ ∈ (0, ∞), for general population covariance. We show that spectral properties for large dimensional correlation matrices are similar to those of large dimensional covariance matrices, for a large class of models studied in random matrix theory. We also derive a MarčenkoPastur type system of equations for the limiting spectral distribution of covariance matrices computed from data with elliptical distributions and generalizations of this family. The motivation for this study comes partly from the possible relevance of such distributional assumptions to problems in econometrics and portfolio optimization, as well as robustness questions for certain classical random matrix results. A mathematical theme of the paper is the important use we make of concentration inequalities.
Quantum permutation groups: a survey
"... Abstract. This is a presentation of recent work on quantum permutation groups. Contains: a short introduction to operator algebras and Hopf algebras; quantum permutation groups, and their basic properties; diagrams, integration formulae, asymptotic laws, matrix models; the hyperoctahedral quantum gr ..."
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Cited by 7 (6 self)
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Abstract. This is a presentation of recent work on quantum permutation groups. Contains: a short introduction to operator algebras and Hopf algebras; quantum permutation groups, and their basic properties; diagrams, integration formulae, asymptotic laws, matrix models; the hyperoctahedral quantum group, free wreath products, quantum automorphism groups of finite graphs, graphs having no quantum symmetry; complex Hadamard matrices, cocycle twists of the symmetric group, quantum groups acting on 4 points; remarks and comments. 1.
On the Law of Addition of Random Matrices
 COMMUNICATIONS IN MATHEMATICAL PHYSICS
, 2000
"... Normalized eigenvalue counting measure of the sum of two Hermitian (or real symmetric) matrices An and Bn rotated independently with respect to each other by the random unitary (or orthogonal) Haar distributed matrix Un (i.e. An + U ∗ n BnUn) is studied in the limit of large matrix order n. Converg ..."
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Cited by 6 (1 self)
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Normalized eigenvalue counting measure of the sum of two Hermitian (or real symmetric) matrices An and Bn rotated independently with respect to each other by the random unitary (or orthogonal) Haar distributed matrix Un (i.e. An + U ∗ n BnUn) is studied in the limit of large matrix order n. Convergence in probability to a limiting nonrandom measure is established. A functional equation for the Stieltjes transform of the limiting measure in terms of limiting eigenvalue measures of An and Bn is obtained and studied.
The spectrum of kernel random matrices
, 2007
"... We place ourselves in the setting of highdimensional statistical inference, where the number of variables p in a dataset of interest is of the same order of magnitude as the number of observations n. We consider the spectrum of certain kernel random matrices, in particular n × n matrices whose (i, ..."
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Cited by 6 (2 self)
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We place ourselves in the setting of highdimensional statistical inference, where the number of variables p in a dataset of interest is of the same order of magnitude as the number of observations n. We consider the spectrum of certain kernel random matrices, in particular n × n matrices whose (i, j)th entry is f(X ′ i Xj/p) or f(‖Xi − Xj ‖ 2 /p), where p is the dimension of the data, and Xi are independent data vectors. Here f is assumed to be a locally smooth function. The study is motivated by questions arising in statistics and computer science, where these matrices are used to perform, among other things, nonlinear versions of principal component analysis. Surprisingly, we show that in highdimensions, and for the models we analyze, the problem becomes essentially linear which is at odds with heuristics sometimes used to justify the usage of these methods. The analysis also highlights certain peculiarities of models widely studied in random matrix theory and raises some questions about their relevance as tools to model highdimensional data encountered in practice. 1
Kalman filtering with intermittent observations: Weak convergence to a stationary distribution. Accepted for publication
 in the IEEE Transactions on Automatic Control with mandatory revisions
, 2009
"... The paper studies the asymptotic behavior of Random Algebraic Riccati Equations (RARE) arising in Kalman filtering when the arrival of the observations is described by a Bernoulli i.i.d. process. We model the RARE as an orderpreserving, strongly sublinear random dynamical system (RDS). Under a suff ..."
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The paper studies the asymptotic behavior of Random Algebraic Riccati Equations (RARE) arising in Kalman filtering when the arrival of the observations is described by a Bernoulli i.i.d. process. We model the RARE as an orderpreserving, strongly sublinear random dynamical system (RDS). Under a sufficient condition, stochastic boundedness, and using a limitset dichotomy result for orderpreserving, strongly sublinear RDS, we establish the asymptotic properties of the RARE: the sequence of random prediction error covariance matrices converges weakly to a unique invariant distribution, whose support exhibits fractal behavior. In particular, this weak convergence holds under broad conditions and even when the observations arrival rate is below the critical probability for mean stability. We apply the weakFeller property of the Markov process governing the RARE to characterize the support of the limiting invariant distribution as the topological closure of a countable set of points, which, in general, is not dense in the set of positive semidefinite matrices. We use the explicit characterization of the support of the invariant distribution and the almost sure ergodicity of the sample paths to easily compute the moments of the invariant distribution. A one dimensional example illustrates that the support is a fractured subset of the nonnegative reals with selfsimilarity properties.
ηseries and a Boolean Bercovici–Pata bijection for bounded ktuples
, 2006
"... Let Dc(k) be the space of (noncommutative) distributions of ktuples of selfadjoint elements in a C∗probability space. On Dc(k) one has an operation ⊞ of free additive convolution, and one can consider the subspace Dinfdiv c (k) of distributions which are infinitely divisible with respect to this ..."
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Cited by 5 (5 self)
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Let Dc(k) be the space of (noncommutative) distributions of ktuples of selfadjoint elements in a C∗probability space. On Dc(k) one has an operation ⊞ of free additive convolution, and one can consider the subspace Dinfdiv c (k) of distributions which are infinitely divisible with respect to this operation. The linearizing transform for ⊞ is the Rtransform (one has R µ⊞ν = Rµ + Rν, ∀ µ, ν ∈ Dc(k)). We prove the equality {Rµ  µ ∈ D infdiv c (k)} = {ηµ  µ ∈ Dc(k)}, (I) where for µ ∈ Dc(k) we denote by ηµ the ηseries associated to µ. (The series ηµ is the counterpart of Rµ in the theory of Boolean convolution. A quick way to define it is by the formula ηµ = Mµ/(1 + Mµ), with Mµ the moment seris of µ.) As a consequence of (I), one can define a bijection B: Dc(k) → Dinfdiv c (k) via the formula R B(µ) = ηµ, ∀ µ ∈ Dc(k). We show that B is a multivariable analogue of a bijection studied by Bercovici and Pata for k = 1, and we prove a theorem about convergence in moments which parallels the BercoviciPata result. On the other hand we prove the formula B(µ ⊠ ν) = B(µ) ⊠ B(ν), (II) (III) with µ, ν considered in a space Dalg(k) ⊇ Dc(k) where the operation of free multiplicative convolution ⊠ always makes sense. An equivalent reformulation of (III) is that η µ⊠ν = ηµ ⋆ ην, ∀ µ, ν ∈ Dalg(k). (IV) This shows that, in a certain sense, ηseries behave in the same way as Rtransforms in connection to the operation of multiplication of free ktuples of noncommutative random variables.
On a remarkable semigroup of homomorphisms with respect to free multiplicative convolution
, 2008
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A simple approach to global regime of the random matrix theory
 Mathematical Results in Statistical Mechanics. Singapore: World Scientific
, 1999
"... Abstract. We discuss a method of the asymptotic computation of moments of the normalized eigenvalue counting measure of random matrices of large order. The method is based on the resolvent identity and on some formulas relating expectations of certain matrix functions and the expectations including ..."
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Cited by 4 (1 self)
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Abstract. We discuss a method of the asymptotic computation of moments of the normalized eigenvalue counting measure of random matrices of large order. The method is based on the resolvent identity and on some formulas relating expectations of certain matrix functions and the expectations including their derivatives or, equivalently, on some simple formulas of the perturbation theory. In the framework of this unique approach we obtain functional equations for the Stieltjes transforms of the limiting normalized eigenvalue counting measure and the bounds for the rate of convergence for the majority known random matrix ensembles. 1.