## A simple invariance theorem (2004)

Citations: | 7 - 1 self |

### BibTeX

@MISC{Chatterjee04asimple,

author = {Sourav Chatterjee},

title = {A simple invariance theorem},

year = {2004}

}

### OpenURL

### Abstract

This is an old article (from May 2004), that will probably not be published, because a much improved paper with new results is in preparation. Still, I decided to put it in the archive because there are some things of interest here (in particular, the section on the S-K model) which will not appear in the new paper. 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 glasses, and maxima of random fields. 1 Introduction and

### Citations

1194 |
The Algebraic Eigenvalue Problem
- Wilkinson
- 1965
(Show Context)
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 =-=[27]-=- pp. 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) and the eigenvalues of ∂A... |

190 |
Spin Glasses: A Challenge for Mathematicians
- Talagrand
- 2003
(Show Context)
Citation Context ...onsequence and interest to physicists, and nowadays, to people in neural networks also. For a detailed discussion of mathematical results about spin glass models and further references, see Talagrand =-=[23]-=-, for instance. 3.2 The Sherrington-Kirkpatrick model The Sherrington-Kirkpatrick (S-K) model, introduced in [22], can be briefly described as follows: For each N ≥ 1 let {JN ij , 1 ≤ i,j ≤ N} be a co... |

189 |
Characteristic vectors of bordered matrices with infinite dimensions
- Wigner
- 1955
(Show Context)
Citation Context ...atrix whose (i,j) th entry is N −1/2 xij if i ≤ j and N −1/2 xji if i > j. If X is a vector of n independent standard Gaussian random variables, then A(X) is a standard Gaussian Wigner matrix. Wigner =-=[26]-=- showed that the LSD for a sequence of standard Gaussian Wigner matrices is the semicircle law, which has density (2π) −1√ 4 − x 2 in [−2,2]. It was later shown that the distribution of the entries do... |

129 |
S.Kirkpatrick, “Solvable model of a spin-glass,” Phys
- Sherrington
- 1975
(Show Context)
Citation Context ... of mathematical results about spin glass models and further references, see Talagrand [23], for instance. 3.2 The Sherrington-Kirkpatrick model The Sherrington-Kirkpatrick (S-K) model, introduced in =-=[22]-=-, can be briefly described as follows: For each N ≥ 1 let {JN ij , 1 ≤ i,j ≤ N} be a collection of i.i.d. N(0,1) random variables. The S-K model assigns a random probability distribution (the Gibbs me... |

102 |
Random Matrices, Second Edition
- Mehta
- 1991
(Show Context)
Citation Context ...s of random matrix theory. For an excellent review of mathematical results known about limiting spectral behaviour and further references, see Bai [2]. For relevance in physics, see the book by Mehta =-=[17]-=-. 2.2 Stieltjes transforms A standard tool for identifying the LSD of a sequence of random matrices is the Stieltjes transform. To cut a long story short, we can say that the ESDs of a sequence {AN } ... |

86 |
Methodologies in spectral analysis of large-dimensional random matrices, a review, Statist
- Bai
- 1999
(Show Context)
Citation Context ...ious kinds of random matrices is one of the main goals of random matrix theory. For an excellent review of mathematical results known about limiting spectral behaviour and further references, see Bai =-=[2]-=-. For relevance in physics, see the book by Mehta [17]. 2.2 Stieltjes transforms A standard tool for identifying the LSD of a sequence of random matrices is the Stieltjes transform. To cut a long stor... |

85 | Noise stability of functions with low influences: invariance and optimality
- Mossel, O’Donnell, et al.
- 2008
(Show Context)
Citation Context ...e to our notice that indeed, there is an old article of Rotar [21] which examines the Lindeberg method polynomial maps in a limiting case. Also, earlier this year, Mossel, O’Donnell, and Oleszkiewicz =-=[18]-=- made some striking applications to problems from computer science and discrete mathematics using the Lindeberg method on polynomials.) We shall derive a very simple extension of Lindeberg’s argument ... |

83 |
The Parisi formula
- Talagrand
- 2005
(Show Context)
Citation Context ...li [12] that the limit 1 lim E(log ZN,J) N→∞ N exists for all β and h. See Talagrand [23] Theorem 2.10.1, p. 140 for a proof. A formula for the limit was conjectured by Parisi and proved by Talagrand =-=[24]-=-. Talagrand ([23] Corollary 2.2.5, p. 32) also proves (in particular) that 1 N (log ZN,J − E log ZN,J) P → 0 for any β and h. Both the above facts were proved under the condition that JN ij are i.i.d.... |

82 | Broken Replica Symmetry Bounds in the Mean Field Spin
- Guerra
- 2003
(Show Context)
Citation Context ...ss model is the configuration which minimizes the Hamiltonian. With β = 1 and h = 0 for simplicity, the energy of the ground state is given by SN(J) = max σ∈ΣN ∑ Jijσiσj. 1≤i<j≤N Guerra and Toninelli =-=[11, 12]-=- proved that N −3/2SN(J) converges almost surely and in average to a deterministic limit if J is a collection of standard Gaussian random variables. It was extended to the case of i.i.d. entries with ... |

70 | The Thermodynamic Limit in Mean Field Spin Glass Models
- Guerra, Toninelli
- 2002
(Show Context)
Citation Context ... β and h are fixed parameters, and ZN,J is the normalising constant. Ideally, the subscripts should include β and h, but we are considering them to be fixed. It has been shown by Guerra and Toninelli =-=[12]-=- that the limit 1 lim E(log ZN,J) N→∞ N exists for all β and h. See Talagrand [23] Theorem 2.10.1, p. 140 for a proof. A formula for the limit was conjectured by Parisi and proved by Talagrand [24]. T... |

53 |
Probabilities on Algebraic Structures
- Grenander
- 1963
(Show Context)
Citation Context ...n [−2,2]. It was later shown that the distribution of the entries do not play a significant role: convergence to the semicircle law would hold under more general conditions (Cf. Arnold [1], Grenander =-=[10]-=- and Bai [2]). The weakest known condition under which the convergence to semicircle law holds was given by Pastur [20]. It is claimed that the condition was shown to be necessary by Girko [9]. For a ... |

44 |
On the asymptotic distribution of eigenvalues of random matrices
- Arnold
- 1967
(Show Context)
Citation Context ...) −1√ 4 − x 2 in [−2,2]. It was later shown that the distribution of the entries do not play a significant role: convergence to the semicircle law would hold under more general conditions (Cf. Arnold =-=[1]-=-, Grenander [10] and Bai [2]). The weakest known condition under which the convergence to semicircle law holds was given by Pastur [20]. It is claimed that the condition was shown to be necessary by G... |

43 |
The spectrum of random matrices
- Pastur
- 1972
(Show Context)
Citation Context ...micircle law would hold under more general conditions (Cf. Arnold [1], Grenander [10] and Bai [2]). The weakest known condition under which the convergence to semicircle law holds was given by Pastur =-=[20]-=-. It is claimed that the condition was shown to be necessary by Girko [9]. For a detailed exposition, see Bai [2] or Khorunzhy, Khoruzhenko and Pastur [13]. The method of this paper will give an easy ... |

40 | Asymptotic properties of large random matrices with independent entries
- Khorunzhy, Khoruzhenko, et al.
- 1996
(Show Context)
Citation Context ... to semicircle law holds was given by Pastur [20]. It is claimed that the condition was shown to be necessary by Girko [9]. For a detailed exposition, see Bai [2] or Khorunzhy, Khoruzhenko and Pastur =-=[13]-=-. The method of this paper will give an easy way to show the sufficiency of Pastur’s condition. Incidentally, somewhat similar ideas involving derivatives of empirical characteristic functions (instea... |

36 |
An approximation theorem for the Poisson binomial distribution
- Cam
- 1960
(Show Context)
Citation Context ...ds are not so useful. For more information on this topic, see the survey paper [3] and the monograph [19]. (Another possible source is Bergström’s books [4, 5]. It is also worth mentioning that LeCam =-=[14]-=- had a similar idea for Poisson approximation.) The ideas were carefully examined and generalized by Zolotarev [28] through the introduction of the so-called ζ metrics, which we shall not discuss here... |

33 |
Eine neue Herleitung des Exponentialgesetzes in der Wahrscheinlichkeitsrechtnung
- Lindeberg
- 1922
(Show Context)
Citation Context .... We apply the technique to prove results from random matrix theory, spin glasses, and maxima of random fields. 1 Introduction and results J. W. Lindeberg’s elegant proof of the Central Limit Theorem =-=[15, 16]-=-, despite being in the shadow of Fourier analytic methods for a long time, is now well known. It was revived by Trotter [25] and has since been used successfully to derive CLTs in infinite dimensional... |

20 |
Ideal metrics in the problem of approximating the distributions of sums of independent random variables. (Russian.) Teor. Veroyatnost. i Primenen. 22, 449–465. English transl. Theory Probab
- Zolotarev
- 1977
(Show Context)
Citation Context ... possible source is Bergström’s books [4, 5]. It is also worth mentioning that LeCam [14] had a similar idea for Poisson approximation.) The ideas were carefully examined and generalized by Zolotarev =-=[28]-=- through the introduction of the so-called ζ metrics, which we shall not discuss here. However, it seems that the basic method of replacing non-Gaussian random variables by Gaussians one by one and us... |

19 |
The accuracy of Gaussian approximation in Banach spaces
- Bentkus, Götze, et al.
- 1991
(Show Context)
Citation Context ...[25] and has since been used successfully to derive CLTs in infinite dimensional spaces, where the Fourier analytic methods are not so useful. For more information on this topic, see the survey paper =-=[3]-=- and the monograph [19]. (Another possible source is Bergström’s books [4, 5]. It is also worth mentioning that LeCam [14] had a similar idea for Poisson approximation.) The ideas were carefully exami... |

14 |
Limit theorems for polylinear forms
- Rotar
- 1979
(Show Context)
Citation Context ...more general invariance results has been overlooked in the literature. (After the preparation of the initial draft of this article, it came to our notice that indeed, there is an old article of Rotar =-=[21]-=- which examines the Lindeberg method polynomial maps in a limiting case. Also, earlier this year, Mossel, O’Donnell, and Oleszkiewicz [18] made some striking applications to problems from computer sci... |

13 |
On certain limit theorems in the theory of probability
- Erdős, Kac
- 1946
(Show Context)
Citation Context ...σj, where the maximum is taken over all σ ∈ {−1,1} N , is not dependent on the exact distributions of the xij’s. For an immediate application, consider the (very old) question raised by Erdős and Kac =-=[8]-=-: what is the limiting distribution of max1≤j≤n 1 √ n ∑ j i=1 Xi where Xi’s are i.i.d. with mean zero and unit variance? It is now well known that the limiting distribution is the same as that of |Z|,... |

11 |
An elementary proof of the central limit theorem
- Trotter
- 1959
(Show Context)
Citation Context ...nd results J. W. Lindeberg’s elegant proof of the Central Limit Theorem [15, 16], despite being in the shadow of Fourier analytic methods for a long time, is now well known. It was revived by Trotter =-=[25]-=- and has since been used successfully to derive CLTs in infinite dimensional spaces, where the Fourier analytic methods are not so useful. For more information on this topic, see the survey paper [3] ... |

10 |
Weak Convergence of Measures
- Bergström
- 1982
(Show Context)
Citation Context ...sional spaces, where the Fourier analytic methods are not so useful. For more information on this topic, see the survey paper [3] and the monograph [19]. (Another possible source is Bergström’s books =-=[4, 5]-=-. It is also worth mentioning that LeCam [14] had a similar idea for Poisson approximation.) The ideas were carefully examined and generalized by Zolotarev [28] through the introduction of the so-call... |

10 |
Spectral Theory of Random Matrices
- Girko
- 1988
(Show Context)
Citation Context ...nander [10] and Bai [2]). The weakest known condition under which the convergence to semicircle law holds was given by Pastur [20]. It is claimed that the condition was shown to be necessary by Girko =-=[9]-=-. For a detailed exposition, see Bai [2] or Khorunzhy, Khoruzhenko and Pastur [13]. The method of this paper will give an easy way to show the sufficiency of Pastur’s condition. Incidentally, somewhat... |

7 | Universality in Sherrington-Kirkpatrick’s spin glass model
- Carmona
- 2006
(Show Context)
Citation Context ...n fact, the rigorous proofs involve the use of intricate properties of Gaussian random variables. Recently, in a paper which was archived at a time when this article was being written, Carmona and Hu =-=[6]-=- have proved that the limit will exist and be the same when Jij are i.i.d. with zero mean, unit variance and finite third moment. Their technique may be extended to the case of independent variables w... |

2 |
Approximation theory in the Central Limit Theorem. Exact Results in Banach Spaces
- Paulauskus, Račkauskus
- 1989
(Show Context)
Citation Context ...n used successfully to derive CLTs in infinite dimensional spaces, where the Fourier analytic methods are not so useful. For more information on this topic, see the survey paper [3] and the monograph =-=[19]-=-. (Another possible source is Bergström’s books [4, 5]. It is also worth mentioning that LeCam [14] had a similar idea for Poisson approximation.) The ideas were carefully examined and generalized by ... |

1 |
Limit theorems for convolutions. Almqvist & Wiksell, StockholmG¨teborg-Uppsala
- Bergström
- 1963
(Show Context)
Citation Context ...sional spaces, where the Fourier analytic methods are not so useful. For more information on this topic, see the survey paper [3] and the monograph [19]. (Another possible source is Bergström’s books =-=[4, 5]-=-. It is also worth mentioning that LeCam [14] had a similar idea for Poisson approximation.) The ideas were carefully examined and generalized by Zolotarev [28] through the introduction of the so-call... |

1 |
Über das Exponentialgesetzes in der
- Lindeberg
- 1920
(Show Context)
Citation Context .... We apply the technique to prove results from random matrix theory, spin glasses, and maxima of random fields. 1 Introduction and results J. W. Lindeberg’s elegant proof of the Central Limit Theorem =-=[15, 16]-=-, despite being in the shadow of Fourier analytic methods for a long time, is now well known. It was revived by Trotter [25] and has since been used successfully to derive CLTs in infinite dimensional... |