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24
Stein’s method on Wiener chaos
 Probab. Theory Relat. Fields
, 2009
"... Abstract: We combine Malliavin calculus with Stein’s method, in order to derive explicit bounds in the Gaussian and Gamma approximations of random variables in a fixed Wiener chaos of a general Gaussian process. Our approach generalizes, refines and unifies the central and noncentral limit theorems ..."
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Cited by 33 (25 self)
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Abstract: We combine Malliavin calculus with Stein’s method, in order to derive explicit bounds in the Gaussian and Gamma approximations of random variables in a fixed Wiener chaos of a general Gaussian process. Our approach generalizes, refines and unifies the central and noncentral limit theorems for multiple WienerItô integrals recently proved (in several papers, from 2005 to 2007) by Nourdin, Nualart, OrtizLatorre, Peccati and Tudor. We apply our techniques to prove BerryEsséen bounds in the BreuerMajor CLT for subordinated functionals of fractional Brownian motion. By using the wellknown Mehler’s formula for OrnsteinUhlenbeck semigroups, we also recover a technical result recently proved by Chatterjee, concerning the Gaussian approximation of functionals of finitedimensional Gaussian vectors.
A new method of normal approximation
, 2006
"... Abstract. We introduce a new version of Stein’s method that reduces a large class of normal approximation problems to variance bounding exercises, thus making a connection between central limit theorems and concentration of measure. Unlike Skorokhod embeddings, the object whose variance has to be bo ..."
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Cited by 23 (4 self)
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Abstract. We introduce a new version of Stein’s method that reduces a large class of normal approximation problems to variance bounding exercises, thus making a connection between central limit theorems and concentration of measure. Unlike Skorokhod embeddings, the object whose variance has to be bounded has an explicit formula that makes it possible to carry out the program more easily. As an application, we derive a general CLT for functions that are obtained as combinations of many local contributions, where the definition of ‘local ’ itself depends on the data. Several examples are given, including the solution to a nearestneighbor CLT problem posed by Peter Bickel. 1.
Multivariate normal approximation using Stein’s method and Malliavin calculus
 Ann. I.H.P
, 2010
"... Abstract: We combine Stein’s method with Malliavin calculus in order to obtain explicit bounds in the multidimensional normal approximation (in the Wasserstein distance) of functionals of Gaussian fields. Our results generalize and refine the main findings by Peccati and Tudor (2005), Nualart and Or ..."
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Cited by 19 (13 self)
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Abstract: We combine Stein’s method with Malliavin calculus in order to obtain explicit bounds in the multidimensional normal approximation (in the Wasserstein distance) of functionals of Gaussian fields. Our results generalize and refine the main findings by Peccati and Tudor (2005), Nualart and OrtizLatorre (2007), Peccati (2007) and Nourdin and Peccati (2007b, 2008); in particular, they apply to approximations by means of Gaussian vectors with an arbitrary, positive definite covariance matrix. Among several examples, we provide an application to a functional version of the BreuerMajor CLT for fields subordinated to a fractional Brownian motion.
Density estimates and concentration inequalities with Malliavin calculus
"... We show how to use the Malliavin calculus to obtain density estimates of the law of general centered random variables. In particular, under a nondegeneracy condition, we prove and use a new formula for the density ρ of a random variable Z which is measurable and di erentiable with respect to a give ..."
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Cited by 13 (8 self)
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We show how to use the Malliavin calculus to obtain density estimates of the law of general centered random variables. In particular, under a nondegeneracy condition, we prove and use a new formula for the density ρ of a random variable Z which is measurable and di erentiable with respect to a given isonormal Gaussian process. Among other results, we apply our techniques to bound the density of the maximum of a general Gaussian process from above and below; several new results ensue, including improvements on the socalled BorellSudakov inequality. We then explain what can be done when one is only interested in or capable of deriving concentration inequalities, i.e. tail bounds from above or below but not necessarily both simultaneously.
Density formula and concentration inequalities with Malliavin calculus
"... We show how to use the Malliavin calculus to obtain a new exact formula for the density ρ of the law of any random variable Z which is measurable and di erentiable with respect to a given isonormal Gaussian process. The main advantage of this formula is that it does not refer to the divergence opera ..."
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Cited by 12 (6 self)
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We show how to use the Malliavin calculus to obtain a new exact formula for the density ρ of the law of any random variable Z which is measurable and di erentiable with respect to a given isonormal Gaussian process. The main advantage of this formula is that it does not refer to the divergence operator (dual of the Malliavin derivative). In particular, density lower bounds can be obtained in some instances. Among several examples, we provide an application to the (centered) maximum of a general Gaussian process, thus extending a formula recently used by Chatterjee [4]. We also explain how to derive concentration inequalities for Z in our framework.
Central limit theorem for linear eigenvalue statistics of random matrices with . . .
, 2009
"... We consider n × n real symmetric and Hermitian Wigner random matrices n −1/2 W with independent (modulo symmetry condition) entries and the (null) sample covariance matrices n −1 X ∗ X with independent entries of m × n matrix X. Assuming first that the 4th cumulant (excess) κ4 of entries of W and X ..."
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Cited by 9 (0 self)
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We consider n × n real symmetric and Hermitian Wigner random matrices n −1/2 W with independent (modulo symmetry condition) entries and the (null) sample covariance matrices n −1 X ∗ X with independent entries of m × n matrix X. Assuming first that the 4th cumulant (excess) κ4 of entries of W and X is zero and that their 4th moments satisfy a Lindeberg type condition, we prove that linear statistics of eigenvalues of the above matrices satisfy the central limit theorem (CLT) as n → ∞, m → ∞, m/n → c ∈ [0, ∞) with the same variance as for Gaussian matrices if the test functions of statistics are smooth enough (essentially of the class C 5). This is done by using a simple “interpolation trick ” from the known results for the Gaussian matrices and the integration by parts, presented in the form of certain differentiation formulas. Then, by using a more elaborated version of the techniques, we prove the CLT in the case of nonzero excess of entries again for essentially C 5 test function. Here the variance of statistics contains an additional term proportional to κ4. The proofs of all limit theorems follow essentially the same scheme.
Imaging schemes for perfectly conducting cracks
 Reconstruction of Small Inhomogeneities from Boundary Measurements, Lecture Notes in Mathematics
, 2004
"... Abstract. We consider the problem of locating perfectly conducting cracks and estimating their geometric features from multistatic response matrix measurements at a single or multiple frequencies. A main objective is to design specific crack detection rules and to analyze their receiver operating c ..."
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Cited by 6 (4 self)
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Abstract. We consider the problem of locating perfectly conducting cracks and estimating their geometric features from multistatic response matrix measurements at a single or multiple frequencies. A main objective is to design specific crack detection rules and to analyze their receiver operating characteristics and the associated signaltonoise ratios. In this paper we introduce an analytic framework that uses asymptotic expansions which are uniform with respect to the wavelengthtocrack size ratio in combination with a hypothesis test based formulation to construct specific procedures for detection of perfectly conducting cracks. A central ingredient in our approach is the use of random matrix theory to characterize the signal space associated with the multistatic response matrix measurements. We present numerical experiments to illustrate some of our main findings.
A new approach for mutual information analysis of large dimensional multiantenna chennels
 4004, 2008. FOR CERTAIN STATISTICS OF GRAM RANDOM MATRICES 41
"... This paper adresses the behaviour of the mutual information of correlated MIMO Rayleigh channels when the numbers of transmit and receive antennas converge to + ∞ at the same rate. Using a new and simple approach based on PoincaréNash inequality and on an integration by parts formula, it is rigorou ..."
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Cited by 6 (5 self)
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This paper adresses the behaviour of the mutual information of correlated MIMO Rayleigh channels when the numbers of transmit and receive antennas converge to + ∞ at the same rate. Using a new and simple approach based on PoincaréNash inequality and on an integration by parts formula, it is rigorously established that the mutual information when properly centered and rescaled converges to a Gaussian random variable whose mean and variance are evaluated. These results confirm previous evaluations based on the powerful but non rigorous replica method. It is believed that the tools that are used in this paper are simple, robust, and of interest for the communications engineering community.
Second order Poincaré inequalities and CLTs on Wiener space
"... Abstract: We prove in nitedimensional second order Poincaré inequalities on Wiener space, thus closing a circle of ideas linking limit theorems for functionals of Gaussian elds, Stein's method and Malliavin calculus. We provide two applications: (i) to a new second order characterization of CLTs on ..."
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Cited by 6 (6 self)
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Abstract: We prove in nitedimensional second order Poincaré inequalities on Wiener space, thus closing a circle of ideas linking limit theorems for functionals of Gaussian elds, Stein's method and Malliavin calculus. We provide two applications: (i) to a new second order characterization of CLTs on a xed Wiener chaos, and (ii) to linear functionals of Gaussiansubordinated elds. Key words: central limit theorems; isonormal Gaussian processes; linear functionals; multiple integrals; second order Poincaré inequalities; Stein's method; Wiener chaos 2000 Mathematics Subject Classi cation: 60F05; 60G15; 60H07 1
A new approach to strong embeddings
, 2007
"... Abstract. We revisit strong approximation theory from a new perspective, culminating in a proof of the KomlósMajorTusnády embedding theorem for the simple random walk. The proof is almost entirely based on a series of soft arguments and easy inequalities. The new technique, inspired by Stein’s met ..."
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Cited by 1 (0 self)
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Abstract. We revisit strong approximation theory from a new perspective, culminating in a proof of the KomlósMajorTusnády embedding theorem for the simple random walk. The proof is almost entirely based on a series of soft arguments and easy inequalities. The new technique, inspired by Stein’s method of normal approximation, is applicable to any setting where Stein’s method works. In particular, one can hope to take it far beyond sums of independent random variables. 1.