## A generalization of the Lindeberg principle (2006)

Venue: | Annals Probab |

Citations: | 12 - 0 self |

### BibTeX

@ARTICLE{Chatterjee06ageneralization,

author = {Sourav Chatterjee},

title = {A generalization of the Lindeberg principle},

journal = {Annals Probab},

year = {2006},

pages = {2061--2076}

}

### OpenURL

### Abstract

We present a generalization of Lindeberg’s method of proving the central limit theorem to encompass general smooth functions (instead of just sums) and dependent random variables. The technique is then used to obtain an invariance result for smooth functions of exchangeable random variables. As an illustrative application of this theorem, we then establish “convergence to Wigner’s law ” for eigenspectra of matrices with exchangeable random entries. 1 Introduction and

### Citations

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Citation Context ...l, with the diagonal elements being the eigenvalues of B, and then apply the second property. The above facts are standard, and may be looked up in any standard text on matrix analysis. See Wilkinson =-=[18]-=- pages 55–58, for example. Now, it is easy to see that G and the derivatives of A are all normal matrices. Moreover, the eigenvalues of G are bounded by |v| −1 , where v = Im z. Thus, by the propertie... |

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Citation Context ... 4An application: Wigner’s law for exchangeable random variables. The spectral theory of large dimensional random matrices is a growing area of contemporary research in probability. Wigner’s theorem =-=[17]-=- (to be described below) was one of the earliest results in this field. We will demonstrate how to extend Wigner’s theorem, which was originally proved for matrices with i.i.d. random entries, to the ... |

89 | Noise stability of functions with low influences: invariance and optimality
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Citation Context ...not only for sums, but for more general smooth functions as well. Comparison of f(X1,...,Xn) and f(Y1,... ,Yn) for polynomial f has been examined by Rotar [16] and Mossel, O’Donnell, and Oleszkiewicz =-=[13]-=-, and for general smooth f with bounded derivatives by Chatterjee [5]. In [5], it is shown how to apply the method to establish universality in physical models, including the Sherrington-Kirkpatrick m... |

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Citation Context ...on function. We will denote the c.d.f. for the ESD of a matrix A by FA. For an excellent review of mathematical results known about limiting spectral behaviour and further references, we refer to Bai =-=[2]-=-. For relevance in physics, the book by Mehta [10] is an essential reference. Wigner matrices. A random Wigner matrix of order N is an N × N real symmetric matrix with independent entries on and above... |

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Citation Context ... techniques. Before moving on to applications, we will now briefly mention the interpretation of Theorem 1.2 in the light of de Finetti’s theorem [4] and its finite version due to Diaconis & Freedman =-=[5]-=-. According to the general de Finetti paradigm in the finite setting, small subcollections of {X1,... ,Xn} behave approximately like collections of i.i.d. random variables with mean ˆµ and variance ˆσ... |

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Citation Context ...do not play a significant role; convergence to the semicircle law would hold under more general conditions (Cf. Arnold [1], Grenander [7] and Bai [2]). The weakest known condition was given by Pastur =-=[12]-=-. It is claimed that the condition was shown to be necessary by Girko [6]. However, all conditions require independence of the entries on and above the diagonal. For a detailed exposition, see Bai [2]... |

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Citation Context ... shown to be necessary by Girko [6]. However, all conditions require independence of the entries on and above the diagonal. For a detailed exposition, see Bai [2] or Khorunzhy, Khoruzhenko and Pastur =-=[8]-=-. We will extend Wigner’s theorem in a different direction: Instead of the original i.i.d. assump5tion, we will assume that the entries above the diagonal are exchangeable. Of course, we will not get... |

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33 |
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Citation Context ... we then establish “convergence to Wigner’s law” for eigenspectra of matrices with exchangeable random entries. 1 Introduction and results J. W. Lindeberg’s elegant proof of the Central Limit Theorem =-=[9]-=-, despite being in the shadow of Fourier analytic methods for a long time, is now well-known. It was revived by Trotter [16] and has since been used successfully to derive CLTs in infinite dimensional... |

20 |
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Citation Context .... Let us now say a bit about the condition of boundedness of third derivatives. The implications of this condition have been inspected in detail in the context of convolutions by Zolotarev (see, e.g. =-=[19]-=-) and other authors. Zolotarev defines the ζ3-metric on the space of distributions as follows: ζ3(F,G) = sup f | ∫ fdF − ∫ fdG|, where the sup is taken over all f with third derivative bounded by 1. T... |

17 |
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Citation Context ... [18] that the results in [5] can be used to give an immediate proof of the universality of last passage percolation in thin rectangles (originally a result of Baik & Suidan [3] and Bodineau & Martin =-=[4]-=-). Developed independently, the Mossel et al. paper [13] is another repository of very striking modern applications of this very old idea. A closer examination of Lindeberg’s method reveals that there... |

14 |
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(Show Context)
Citation Context ... becomes clear that the same principle works not only for sums, but for more general smooth functions as well. Comparison of f(X1,...,Xn) and f(Y1,... ,Yn) for polynomial f has been examined by Rotar =-=[14]-=- and for general smooth f with bounded derivatives by Chatterjee [3]. In [3], the author has an application which shows how to apply the method to establish universality in physical models, including ... |

14 | A GUE central limit theorem and universality of directed first and last passage site percolation
- Baik, Suidan
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Citation Context ... observed by Toufic Suidan [18] that the results in [5] can be used to give an immediate proof of the universality of last passage percolation in thin rectangles (originally a result of Baik & Suidan =-=[3]-=- and Bodineau & Martin [4]). Developed independently, the Mossel et al. paper [13] is another repository of very striking modern applications of this very old idea. A closer examination of Lindeberg’s... |

10 |
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(Show Context)
Citation Context ...ld under more general conditions (Cf. Arnold [1], Grenander [7] and Bai [2]). The weakest known condition was given by Pastur [12]. It is claimed that the condition was shown to be necessary by Girko =-=[6]-=-. However, all conditions require independence of the entries on and above the diagonal. For a detailed exposition, see Bai [2] or Khorunzhy, Khoruzhenko and Pastur [8]. We will extend Wigner’s theore... |

10 |
1959]: "An Elementary Proof of the Central Limit Theorem
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Citation Context ...ion and results J. W. Lindeberg’s elegant proof of the Central Limit Theorem [9], despite being in the shadow of Fourier analytic methods for a long time, is now well-known. It was revived by Trotter =-=[16]-=- and has since been used successfully to derive CLTs in infinite dimensional spaces, where the Fourier analytic methods are not so useful. While the original Lindeberg method and most of its extension... |

9 | Semicircle law and freeness for random matrices with symmetries or correlations
- Schenker, Schulz-Baldes
(Show Context)
Citation Context ... is claimed that the condition was shown to be necessary by Girko [9]. Although most conditions require independence of the entries on and above the diagonal, there have been some advances (e.g. [7], =-=[17]-=-) allowing certain kinds of dependence. However, none of these cover the case of exchangeable entries. For a detailed exposition of the key results about the spectra of Wigner matrices and other resul... |

7 | A simple invariance theorem - Chatterjee - 2005 |

3 |
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Citation Context ...]. It is claimed that the condition was shown to be necessary by Girko [9]. Although most conditions require independence of the entries on and above the diagonal, there have been some advances (e.g. =-=[7]-=-, [17]) allowing certain kinds of dependence. However, none of these cover the case of exchangeable entries. For a detailed exposition of the key results about the spectra of Wigner matrices and other... |

3 |
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(Show Context)
Citation Context ...e [5]. In [5], it is shown how to apply the method to establish universality in physical models, including the Sherrington-Kirkpatrick model of spin glasses. It was recently observed by Toufic Suidan =-=[18]-=- that the results in [5] can be used to give an immediate proof of the universality of last passage percolation in thin rectangles (originally a result of Baik & Suidan [3] and Bodineau & Martin [4]).... |

2 |
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(Show Context)
Citation Context .... This will be formalized using gaussian interpolation techniques. Before moving on to applications, we will now briefly mention the interpretation of Theorem 1.2 in the light of de Finetti’s theorem =-=[4]-=- and its finite version due to Diaconis & Freedman [5]. According to the general de Finetti paradigm in the finite setting, small subcollections of {X1,... ,Xn} behave approximately like collections o... |

2 |
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Citation Context ...ful. While the original Lindeberg method and most of its extensions compare the distributions of convolutions in great generality (the history of which is irrelevant to our discussion, so we refer to =-=[13]-=- for details), it soon becomes clear that the same principle works not only for sums, but for more general smooth functions as well. Comparison of f(X1,...,Xn) and f(Y1,... ,Yn) for polynomial f has b... |

1 |
A simple invariance theorem. Available at http://arxiv.org/math.PR/0508213
- Chatterjee
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(Show Context)
Citation Context ...arison of f(X1,...,Xn) and f(Y1,... ,Yn) for polynomial f has been examined by Rotar [16] and Mossel, O’Donnell, and Oleszkiewicz [13], and for general smooth f with bounded derivatives by Chatterjee =-=[5]-=-. In [5], it is shown how to apply the method to establish universality in physical models, including the Sherrington-Kirkpatrick model of spin glasses. It was recently observed by Toufic Suidan [18] ... |