A generalization of the Lindeberg principle

By Sourav Chatterjee. 2005

197 citations recorded by the source corpus.

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.

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Citations in the corpus

The Isotropic Semicircle Law and Deformation of Wigner Matrices (2011, methodology): “It uses a Lindeberg replacement strategy, which previously appeared in the context of random matrix theory in [14, 33, 34].”
A short survey of Stein's method (2014, methodology): “I think I may have been the first one to realize in [17, 19] that the Lindeberg method applies to general functions (and not just sums and polynomials), with a potentially wide range of interesting applications.”
Anisotropic local laws for random matrices (2014, background): “1007/s00440-015-0628-6 [10] Chatterjee, S, A generalization of the lindeberg principle, Ann.”
Edge universality of correlation matrices (2011, methodology): “In previous works mentioned earlier, the authors use a Lindeberg replacement strategy, as in [2, 27].”
On the limiting spectral distribution for a large class of symmetric random matrices with correlated entries (2015, methodology): “(See the equalities (20) and (21) in [8] together with the computations on pages 2074-2075).”
The Elliptic Law (2012, methodology): “Recently, this technique has also been applied to study random matrices with exchangeable or independent entries (see, for example, [5, 47] and references therein).”
Universality of local spectral statistics of random matrices (2011, methodology): “The idea of applying Lindeberg’s method in random matrices was recently used by Chatterjee [13] for comparing the traces of the Green functions; the idea was also used by Tao and Vu [67] in the context of comparing individual eigenvalue distributions.”
Random matrix theory in statistics: A review (2014, methodology): “Further research proving similar results under a variety of settings has been carried out by Anderson and Zeitouni (2006), Bai and Silverstein (2004), Bai and Yao (2005), Chatterjee (2006), Johansson (1998), Lytova and Pastur (2009) and Shcherbina (2011), among others.”
Random matrices: Universality of local spectral statistics of non-Hermitian matrices (2012, methodology): “ the resolvent of Hermitian matrices with respect to permutation in just one or two entries, in order to perform swapping arguments. Such swapping arguments were introduced to random matrix theory in [11], and ﬁrst applied to establish universality results for local spectral statistics in [55]. In [”
Alternative asymptotics for cointegration tests in large VARs (2016, methodology): “If the data generating process is pure random walk with zero initial values, then (8) can be rewritten as”
Sparse regular random graphs: Spectral density and eigenvectors (2009, methodology): “Acknowledgements It is our pleasure to thank Chris Burdzy, Sourav Chatterjee, Manju Krishnapur, Nati Linial, and Sasha Soshnikov for very useful discussions.”