Fluctuations of eigenvalues and second order Poincaré inequalities

By Sourav Chatterjee. 2007

235 citations recorded by the source corpus.

Linear statistics of eigenvalues in many familiar classes of random matrices are known to obey gaussian central limit theorems. The proofs of such results are usually rather difficult, involving hard computations specific to the model in question. In this article we attempt to formulate a unified technique for deriving such results via relatively soft arguments. In the process, we introduce a notion of ‘second order Poincaré inequalities’: just as ordinary Poincaré inequalities give variance bounds, second order Poincaré inequalities give central limit theorems. The proof of the main result employs Stein’s method of normal approximation. A number of examples are worked out, some of which are new. One of the new results is a CLT for the spectrum of gaussian Toeplitz matrices.

Subfields: Discrete Mathematics and Combinatorics, Mathematical Physics, Statistics and Probability

View Paper: https://www.semanticscholar.org/paper/8ebb59fb7d11958175f20d4359e6f0a7e73fcc4f

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Citations in the corpus

A NEW METHOD OF NORMAL APPROXIMATION (2006, methodology): “ limitations of these existing techniques. The power of the method is demonstrated through several applications, mainly geometrical in nature, that are diﬃcult to do otherwise. In the related article [11], we give some applications to random matrices. The paper is organized as follows. Section 2 con”
High-dimensional central limit theorems by Stein’s method (2020, methodology): “We first consider random vectors that have a Stein kernel, which was defined in Ledoux, Nourdin and Peccati (2015) and used implicitly in, for example, Chatterjee (2009) and Nourdin and Peccati (2009) (see also Lecture VI of Stein (1986)).”
Multivariate normal approximation using Stein's method and Malliavin calculus (2008, background): “The following result has been proved by Chatterjee in [3], in the context of limit theorems for linear statistics of eigenvalues of random matrices.”
Normal approximation on Poisson spaces: Mehler’s formula, second order Poincaré inequalities and stabilization (2014, methodology): “2) Motivated by problems in random matrix theory, Chatterjee [9] has extended (1.”
On Stein's Method for Infinitely Divisible Laws with Finite First Moment (2017, background): “several quantitative convergence results follow). (iii) As ﬁrst noticed in [41], the quantity 1 2 Xn k=1 Z+N −N k Kν(t,N) n −K(t,N) dt, is the L1 analog of the L2 Stein discrepancy considered e.g. in [7, 8, 26, 21]. As such, it seems to be the appropriate Stein quantity to deal with sums of independ”
Lectures on the Poisson Process (2017, background): “3 was proved in [89] after Chatterjee [21] proved a corresponding result for Gaussian vectors.”
Quantitative Stochastic Homogenization and Large-Scale Regularity (2017, methodology): “Let U ⊆ Rd be a bounded Lipschitz domain and p ∈ [32 ,3].”
Regularity conditions in the CLT for linear eigenvalue statistics of Wigner matrices (2012, background): “In recent years, several authors have derived central limit theorems for centred linear eigenvalue statistics of Wigner matrices with various hypotheses, see [1], [5], [7], [6], [9], [20], [21], [26], [29], [31], [32].”
Second order Poincaré inequalities and CLTs on Wiener space (2008, background): “In the recent paper [4], Chatterjee has pointed out that if one focuses also on the d × d Hessian matrix Hess f , and not only on ∇f , then one can state an inequality assessing the total variation distance (see Section 3.”
Variance estimation in high-dimensional linear models (2014, methodology): “[Theorem 2.2, (Chatterjee, 2009)] Let v = (v1, . . . , vm) ∼ N(0,Ψ).”
Stein couplings for normal approximation (2010, background): “In the case of local dependence, these quantities can now be bounded relatively easily; see Chen and Shao (2004). If truncated fourth moments are to be avoided and no boundedness can be assumed, it seems that more structure is needed in the coupling.”
CENTRAL LIMIT THEOREM FOR LINEAR EIGENVALUE STATISTICS OF RANDOM MATRICES WITH INDEPENDENT ENTRIES (2008, background): “In the last decade a number of results on the CLT for linear eigenvalue statistics of various classes of random matrices has been obtained (see [2, 5, 7, 8, 10, 11, 13, 15, 16, 17, 18, 28, 29, 30] and [31] and the references therein).”
Stein kernels and moment maps (2018, background): “2 below for those maps, so we would only get a useful quantitative estimate in the uniformly convex setting, and due to the particular form of the construction of [10], even in the latter setting the quantitative estimates would get worse.”
A short survey of Stein's method (2014, methodology): “Indeed, it may be said that the whole thriving area of Stein’s method in Malliavin calculus, pioneered by Nourdin and Peccati [60], is an “abstractification” of the ideas contained in [21] and [22].”