A generalization of the Lindeberg principle

@article{Chatterjee2006AGO,
  title={A generalization of the Lindeberg principle},
  author={Sourav Chatterjee},
  journal={Annals of Probability},
  year={2006},
  volume={34},
  pages={2061-2076}
}
  • S. Chatterjee
  • Published 26 August 2005
  • Mathematics
  • Annals of Probability
We generalize Lindeberg’s proof of the central limit theorem to an invariance principle for arbitrary smooth functions of independent and weakly dependent random variables. The result is applied to get a similar theorem for smooth functions of exchangeable random variables. This theorem allows us to identify, for the first time, the limiting spectral distributions of Wigner matrices with exchangeable entries. 
Approximation to stable law by the Lindeberg principle
Lindeberg’s Method for Moderate Deviations and Random Summation
We apply Lindeberg’s method, invented to prove a central limit theorem, to analyze the moderate deviations around such a central limit theorem. In particular, we will show moderate deviation
Limiting Spectral Distribution of Large Sample Covariance Matrices Associated with a Class of Stationary Processes
In this paper, we derive an extension of the Marc̆enko–Pastur theorem to a large class of weak dependent sequences of real-valued random variables having only moment of order 2. Under a mild
On the universality of spectral limit for random matrices with martingale differences entries
For a class of symmetric random matrices whose entries are martingale differences adapted to an increasing filtration, we prove that under a Lindeberg-like condition, the empirical spectral
The LIL for U-Statistics in Hilbert Spaces
Abstract We give necessary and sufficient conditions for the (bounded) law of the iterated logarithm for U-statistics in Hilbert spaces. As a tool we also develop moment and tail estimates for
Comparison theorem for some extremal eigenvalue statistics
We introduce a method for the comparison of some extremal eigenvalue statistics of random matrices. For example, it allows one to compare the maximal eigenvalue gap in the bulk of two generalized
Universality of Limiting Spectral Distribution Under Projective Criteria
This paper has double scope. In the first part we study the limiting empirical spectral distribution of a n × n symmetric matrix with dependent entries. For a class of generalized martingales we show
Limiting spectral distribution for Wigner matrices with dependent entries
In this article we show the existence of limiting spectral distribution of a symmetric random matrix whose entries come from a stationary Gaussian process with covariances satisfying a summability
General Approach to Constructing Non-Asymptotic Bounds
In this chapter, we consider asymptotic expansions for a class of sequences of symmetric functions of many variables. It implies a general approach to get the non-asymptotic bounds for accuracy of
...
1
2
3
4
5
...

References

SHOWING 1-10 OF 26 REFERENCES
A simple invariance theorem
We present a simple extension of Lindeberg's argument for the Central Limit Theorem to get a general invariance result. We apply the technique to prove results from random matrix theory, spin
Limit theorems for polylinear forms
A GUE central limit theorem and universality of directed first andlast passage site percolation
We prove a GUE central limit theorem for random variables with finite fourth moment. We apply this theorem to prove that the directed first and last passage percolation problems in thin rectangles
A remark on a theorem of Chatterjee and last passage percolation
In this paper we prove universality of random matrix fluctuations of the last passage time of last passage percolation (LPP) in thin rectangles. The proof is a simple corollary of a theorem of
The Spectrum of Random Matrices
The Frobenius eigenvector of a positive square matrix is obtained by iterating the multiplication of an arbitrary positive vector by the matrix. Bródy (1997) noticed that, when the entries of the
Semicircle law and freeness for random matrices with symmetries or correlations
For a class of random matrix ensembles with corre- lated matrix elements, it is shown that the density of states is given by the Wigner semi-circle law. This is applied to effective Hamiltonians
A Universality Property for Last-Passage Percolation Paths Close to the Axis
We consider a last-passage directed percolation model in $Z_+^2$, with i.i.d. weights whose common distribution has a finite $(2+p)$th moment. We study the fluctuations of the passage time from the
On the Distribution of the Roots of Certain Symmetric Matrices
TLDR
The distribution law obtained before' for a very special set of matrices is valid for much more general sets of real symmetric matrices of very high dimensionality.
Asymptotic distribution of eigenvalues of random matrices
The impetus for this paper comes mainly from work done in recent years by a number of physicists on a statistical theory of spectra. The book by M. L. Mehta [10] and the collection of reprints edited
Spectral theory of random matrices
CONTENTSIntroductionChapter I. Distribution of the eigenvalues and eigenvectors of random matrices § 1. Polar decomposition of random matrices § 2. Symmetric and Hermitian random matrices § 3.
...
1
2
3
...