## Fluctuations of eigenvalues and second order Poincaré inequalities (2007)

Citations: | 27 - 3 self |

### BibTeX

@MISC{Chatterjee07fluctuationsof,

author = {Sourav Chatterjee},

title = {Fluctuations of eigenvalues and second order Poincaré inequalities},

year = {2007}

}

### Years of Citing Articles

### OpenURL

### Abstract

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 them are new. One of the new results is a CLT for the spectrum of gaussian Toeplitz matrices.

### Citations

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Citation Context ... However, no familiarity with [16] is required here. 1.2. Outline of the proof via Stein’s method. The argument for general g is not as intuitive as for quadratic forms. It begins with Stein’s method =-=[48, 49]-=-: If a random variable W satisfies E(ϕ(W)W) ≈ E(ϕ ′ (W)) for a large class of functions ϕ, then W is approximately standard gaussian. The idea stems from the fact that if W is exactly standard gaussia... |

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Citation Context ...d to grow at the rate o(n2/3 ), instead of remaining fixed. They also get CLTs for Tr(f(An)) for analytic f. Incidentally, for gaussian Wigner matrices, the problem was completely solved by Johansson =-=[34]-=-, characterizing all f for which the CLT holds. In fact, Johansson proved a general result for linear statistics of eigenvalues of random matrices whose entries have a joint density with respect to Le... |

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Citation Context ... However, no familiarity with [16] is required here. 1.2. Outline of the proof via Stein’s method. The argument for general g is not as intuitive as for quadratic forms. It begins with Stein’s method =-=[48, 49]-=-: If a random variable W satisfies E(ϕ(W)W) ≈ E(ϕ ′ (W)) for a large class of functions ϕ, then W is approximately standard gaussian. The idea stems from the fact that if W is exactly standard gaussia... |

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Citation Context ...opic is quite large. To the best of our knowledge, the investigation of central limit theorems for linear statistics of eigenvalues of large dimensional random matrices began with the work of Jonsson =-=[36]-=- on Wishart matrices. The key idea is to express ∑ λk i as ∑ k λi = Tr(A k n ) = ∑ ai1i2ai2i3 · · · aik−1ikaiki1 , i1,i2,...,ik 2000 Mathematics Subject Classification. 60F05, 15A52. Key words and phr... |

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Citation Context ...itary group. An alternative approach, based on Stieltjes transforms, has been developed in Bai and Yao [5] and Bai and Silverstein [6]. This approach has its roots in the semi-rigorous works of Girko =-=[24]-=- and Khorunzhy, Khoruzhenko, and Pastur [38]. Yet another line of attack, via stochastic calculus, was initiated in the work of Cabanal-Duvillard [14]. The ideas were used by Guionnet [26] to prove ce... |

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Citation Context ... with the same mean and variance as W. Then dTV (W,Z) ≤ 2√ 5 σ 2 ( 4c1c2a 2 √ n + 8c3 1 ab n Remarks. (i) It is well known that under mild conditions, λ converges to a finite limit as n → ∞ (see e.g. =-=[4]-=-, Section 2.2.1). Even exponentially decaying tail bounds are available [27]. Thus a and b are generally O(1) in the above bound. (ii) Sinaĭ and Soshnikov ([45], Corollary 1) showed that σ2 converges ... |

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Citation Context ...], we know that E(λn) ≤ C √ log n. Now, it is easy to verify that the map (X0,...,Xn−1) ↦→ λn has Lipschitz constant bounded irrespective of n. By standard gaussian concentration results (e.g. Ledoux =-=[39]-=-, Sections 5.1-5.2), it follows that for any k, E|λn − E(λn)| k ≤ C k/2 k k/2 , where, again, C is a universal constant. Combining with result for E(λn), it follows that for any n and k, E(λ k n) ≤ (C... |

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Citation Context ... Chernoff [18] who used Hermite polynomial expansions. However, such inequalities have been known to analysts for a long time under the name of ‘Hardy inequalities with weights’ (see e.g. Muckenhoupt =-=[41]-=-). ) .6 SOURAV CHATTERJEE We should also mention two other concepts from the existing literature that may be related to this work. The first is the notion of the ‘zerobias transform’ of W, as defined... |

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Citation Context ... √ n + 8c3 1 ab n Remarks. (i) It is well known that under mild conditions, λ converges to a finite limit as n → ∞ (see e.g. [4], Section 2.2.1). Even exponentially decaying tail bounds are available =-=[27]-=-. Thus a and b are generally O(1) in the above bound. (ii) Sinaĭ and Soshnikov ([45], Corollary 1) showed that σ2 converges to a finite limit under certain conditions on f and the distribution of the ... |

61 | Linear functionals of eigenvalues of random matrices
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Citation Context ...n’s proof relies on a delicate analysis of the joint density of the eigenvalues, which is explicitly known for this class of matrices. Another important contribution is the work of Diaconis and Evans =-=[21]-=-, who proved similar results for random unitary matrices. Again, the basic approach relies on the method of moments, but the computations require new ideas because of the lack of independence between ... |

47 | On the characteristic polynomial of a random unitary matrix
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Citation Context ...al ideas, sometimes at varying levels of rigor, come from the papers of Costin and Lebowitz [19], Boutet de Monvel, Pastur and Shcherbina [12], Johansson [33], Keating and Snaith [37], Hughes et. al. =-=[30]-=-, Soshnikov [47], Israelson [31] and Wieand [51]. The recent works of Anderson and Zeitouni [2], Dumitriu and Edelman [22], Rider and Silverstein [43], Rider and Virág [42], Jiang [32], and Hachem et.... |

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Citation Context ... joint convergence of the law of (Tr(An),Tr(A 2 n),... ,Tr(A p n)) to a multivariate normal distribution (where p is fixed). A similar study for Wigner matrices was carried out by Sinaĭ and Soshnikov =-=[45, 46]-=-. A deep and difficult aspect of the Sinaĭ-Soshnikov results is that they get central limit theorems for Tr(A pn n ), where pn is allowed to grow at the rate o(n2/3 ), instead of remaining fixed. They... |

43 |
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Citation Context ...ng some deep connections between symmetric function theory and the unitary group. An alternative approach, based on Stieltjes transforms, has been developed in Bai and Yao [5] and Bai and Silverstein =-=[6]-=-. This approach has its roots in the semi-rigorous works of Girko [24] and Khorunzhy, Khoruzhenko, and Pastur [38]. Yet another line of attack, via stochastic calculus, was initiated in the work of Ca... |

42 | Stein’s Method and zero bias transformation with application to simple random sampling
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Citation Context ... We should also mention two other concepts from the existing literature that may be related to this work. The first is the notion of the ‘zerobias transform’ of W, as defined by Goldstein and Reinert =-=[25]-=-. A random variable W ∗ is called a zerobias transform of W if for all ϕ, we have E(ϕ(W)W) = E(ϕ ′ (W ∗ )). A little consideration shows that our function h is just the density of the law of W ∗ with ... |

40 | Pastur Asymptotic properties of large random matrices with independent entries
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Citation Context ...on Stieltjes transforms, has been developed in Bai and Yao [5] and Bai and Silverstein [6]. This approach has its roots in the semi-rigorous works of Girko [24] and Khorunzhy, Khoruzhenko, and Pastur =-=[38]-=-. Yet another line of attack, via stochastic calculus, was initiated in the work of Cabanal-Duvillard [14]. The ideas were used by Guionnet [26] to prove central limit theorems for certain band matrix... |

37 |
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Citation Context ... [1]. Other influential ideas, sometimes at varying levels of rigor, come from the papers of Costin and Lebowitz [19], Boutet de Monvel, Pastur and Shcherbina [12], Johansson [33], Keating and Snaith =-=[37]-=-, Hughes et. al. [30], Soshnikov [47], Israelson [31] and Wieand [51]. The recent works of Anderson and Zeitouni [2], Dumitriu and Edelman [22], Rider and Silverstein [43], Rider and Virág [42], Jiang... |

34 |
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Citation Context ...xtensions (e.g. [50], [44], [23]), (d) Stein’s method of normal approximation (e.g. [48], [49], [25]), and (e) the big-blocks-smallblocks technique and its modern multidimensional versions (e.g. [9], =-=[3]-=-). For further references — particularly on Stein’s method, which is a cornerstone of our approach — we refer to [16]. Apart from the method of moments, none of the other techniques have been used for... |

34 | Gaussian fluctuation in random matrices
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Citation Context ...trix models were later obtained using combinatorial techniques by Anderson and Zeitouni [1]. Other influential ideas, sometimes at varying levels of rigor, come from the papers of Costin and Lebowitz =-=[19]-=-, Boutet de Monvel, Pastur and Shcherbina [12], Johansson [33], Keating and Snaith [37], Hughes et. al. [30], Soshnikov [47], Israelson [31] and Wieand [51]. The recent works of Anderson and Zeitouni ... |

34 |
A refinement of Wigner’s semicircle law in a neighborhood of the spectrum edge for random symmetric matrices
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Citation Context ... joint convergence of the law of (Tr(An),Tr(A 2 n),... ,Tr(A p n)) to a multivariate normal distribution (where p is fixed). A similar study for Wigner matrices was carried out by Sinaĭ and Soshnikov =-=[45, 46]-=-. A deep and difficult aspect of the Sinaĭ-Soshnikov results is that they get central limit theorems for Tr(A pn n ), where pn is allowed to grow at the rate o(n2/3 ), instead of remaining fixed. They... |

32 | Second order freeness and fluctuations of random matrices: I. Gaussian and Wishart matrices and cyclic Fock spaces
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Citation Context ...Virág [43], Jiang [32], and Hachem et. al. [28, 29] provide several illuminating insights and new results. The recent advances in the theory of second order freeness (introduced by Mingo and Speicher =-=[41]-=-) are also of great interest. In this paper we introduce a result (Theorem 3.1) that may provide a unified ‘soft tool’ for matrices that can be easily expressed as smooth functionsFLUCTUATIONS OF EIG... |

31 |
Sums of functions of nearest neighbor distances, moment bounds, limit theorems and a goodness of fit test
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Citation Context ...ted extensions (e.g. [50], [44], [23]), (d) Stein’s method of normal approximation (e.g. [48], [49], [25]), and (e) the big-blocks-smallblocks technique and its modern multidimensional versions (e.g. =-=[9]-=-, [3]). For further references — particularly on Stein’s method, which is a cornerstone of our approach — we refer to [16]. Apart from the method of moments, none of the other techniques have been use... |

28 | Large deviation upper bounds and central limit theorems for band matrices and non-commutative functionnals of Gaussian large random matrices
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Citation Context ...ks of Girko [24] and Khorunzhy, Khoruzhenko, and Pastur [38]. Yet another line of attack, via stochastic calculus, was initiated in the work of Cabanal-Duvillard [14]. The ideas were used by Guionnet =-=[26]-=- to prove central limit theorems for certain band matrix models. Far reaching results for a very general class of band matrix models were later obtained using combinatorial techniques by Anderson and ... |

27 |
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Citation Context ...orial techniques by Anderson and Zeitouni [1]. Other influential ideas, sometimes at varying levels of rigor, come from the papers of Costin and Lebowitz [19], Boutet de Monvel, Pastur and Shcherbina =-=[12]-=-, Johansson [33], Keating and Snaith [37], Hughes et. al. [30], Soshnikov [47], Israelson [31] and Wieand [51]. The recent works of Anderson and Zeitouni [2], Dumitriu and Edelman [22], Rider and Silv... |

26 |
Fluctuations de la loi empirique des grandes matrices aléatoires
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Citation Context ...has its roots in the semi-rigorous works of Girko [24] and Khorunzhy, Khoruzhenko, and Pastur [38]. Yet another line of attack, via stochastic calculus, was initiated in the work of Cabanal-Duvillard =-=[14]-=-. The ideas were used by Guionnet [26] to prove central limit theorems for certain band matrix models. Far reaching results for a very general class of band matrix models were later obtained using com... |

26 |
A note on an inequality involving the normal distribution
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Citation Context ..., one can view Lemma 1.1 as a generalization of the gaussian Poincaré inequality. Incidentally, the first proof of the gaussian Poincaré inequality in the probability literature is due to H. Chernoff =-=[18]-=- who used Hermite polynomial expansions. However, such inequalities have been known to analysts for a long time under the name of ‘Hardy inequalities with weights’ (see e.g. Muckenhoupt [41]). ) .6 S... |

26 | Gaussian limit for determinantal random point fields
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(Show Context)
Citation Context ...mes at varying levels of rigor, come from the papers of Costin and Lebowitz [19], Boutet de Monvel, Pastur and Shcherbina [12], Johansson [33], Keating and Snaith [37], Hughes et. al. [30], Soshnikov =-=[47]-=-, Israelson [31] and Wieand [51]. The recent works of Anderson and Zeitouni [2], Dumitriu and Edelman [22], Rider and Silverstein [43], Rider and Virág [42], Jiang [32], and Hachem et. al. [28, 29] pr... |

25 | High dimensional statistical inference and random matrices
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Citation Context ...alization, i.e. one does not have to divide by √ n; only centering is enough. Moreover, they have important applications in statistics and other applied areas (see e.g. the recent survey by Johnstone =-=[35]-=-). The literature around the topic is quite large. To the best of our knowledge, the investigation of central limit theorems for linear statistics of eigenvalues of large dimensional random matrices b... |

24 | A new method of normal approximation
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- 2008
(Show Context)
Citation Context ... is involved. We provide a ‘finished product’ in Theorem 3.1 for the convenience of potential future users of the method. A discrete version of this idea is investigated in the author’s earlier paper =-=[16]-=-. However, no familiarity with [16] is required here. 1.2. Outline of the proof via Stein’s method. The argument for general g is not as intuitive as for quadratic forms. It begins with Stein’s method... |

20 |
Eigenvalue distributions of random unitary matrices, Probability Theory and Related Fields 123
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Citation Context ...come from the papers of Costin and Lebowitz [19], Boutet de Monvel, Pastur and Shcherbina [12], Johansson [33], Keating and Snaith [37], Hughes et. al. [30], Soshnikov [47], Israelson [31] and Wieand =-=[51]-=-. The recent works of Anderson and Zeitouni [2], Dumitriu and Edelman [22], Rider and Silverstein [43], Rider and Virág [42], Jiang [32], and Hachem et. al. [28, 29] provide several illuminating insig... |

19 |
A Berry-Esséen Bound for symmetric statistics
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(Show Context)
Citation Context ...mply that g(X) has any special structure, at least from what the author understands. In particular, it does not imply that g(X) breaks up as an approximately additive function as in Hájek projections =-=[50, 23]-=-. It is quite mysterious, at the present level of understanding, as to what causes the gaussianity. (iii) A problem with Theorem 2.2 is that it does not say anything about σ 2 . However, in practice, ... |

17 |
A CLT for a band matrix model. Probab. Theory Related Fields 134
- ANDERSON, ZEITOUNI
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(Show Context)
Citation Context ... central limit theorems for certain band matrix models. Far reaching results for a very general class of band matrix models were later obtained using combinatorial techniques by Anderson and Zeitouni =-=[1]-=-. Other influential ideas, sometimes at varying levels of rigor, come from the papers of Costin and Lebowitz [19], Boutet de Monvel, Pastur and Shcherbina [12], Johansson [33], Keating and Snaith [37]... |

16 |
Variational inequalities with examples and an application to the central limit theorem
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Citation Context ...e, then we can conclude that E(ϕ(W)W) ≈ E(ϕ ′ (W)), and it would follow by Stein’s method that W is approximately standard gaussian. This idea already occurs in the literature on normal approximation =-=[15]-=-. However, it is not at all clear how one can infer facts about h(W) when W is an immensely complex object like a linear statistic of eigenvalues of a Wigner matrix. One of the main contributions of t... |

15 |
A Berry–Esseen bound for functions of independent random variables
- Friedrich
- 1989
(Show Context)
Citation Context ...mply that g(X) has any special structure, at least from what the author understands. In particular, it does not imply that g(X) breaks up as an approximately additive function as in Hájek projections =-=[50, 23]-=-. It is quite mysterious, at the present level of understanding, as to what causes the gaussianity. (iii) A problem with Theorem 2.2 is that it does not say anything about σ 2 . However, in practice, ... |

14 |
Mehrdad Shahshahani, On the eigenvalues of random matrices
- Diaconis
- 1994
(Show Context)
Citation Context ...nitary matrices. Again, the basic approach relies on the method of moments, but the computations require new ideas because of the lack of independence between the matrix entries. However, as shown in =-=[20, 21]-=-, strikingly exact computations are possible in this case by invoking some deep connections between symmetric function theory and the unitary group. An alternative approach, based on Stieltjes transfo... |

13 |
On the convergence of the spectral empirical process of Wigner matrices
- Bai, Yao
- 2005
(Show Context)
Citation Context ...sible in this case by invoking some deep connections between symmetric function theory and the unitary group. An alternative approach, based on Stieltjes transforms, has been developed in Bai and Yao =-=[5]-=- and Bai and Silverstein [6]. This approach has its roots in the semi-rigorous works of Girko [24] and Khorunzhy, Khoruzhenko, and Pastur [38]. Yet another line of attack, via stochastic calculus, was... |

13 | Global spectrum fluctuations for the β-Hermite and β-Laguerre ensembles via matrix models
- Dumitriu, Edelman
(Show Context)
Citation Context ... and Shcherbina [12], Johansson [33], Keating and Snaith [37], Hughes et. al. [30], Soshnikov [47], Israelson [31] and Wieand [51]. The recent works of Anderson and Zeitouni [2], Dumitriu and Edelman =-=[22]-=-, Rider and Silverstein [43], Rider and Virág [42], Jiang [32], and Hachem et. al. [28, 29] provide several illuminating insights and new results. In this paper we introduce a result (Theorem 3.1) tha... |

13 | Pastur, “A new approach for capacity analysis of large dimensional multiantenna channels
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(Show Context)
Citation Context ...hnikov [47], Israelson [31] and Wieand [51]. The recent works of Anderson and Zeitouni [2], Dumitriu and Edelman [22], Rider and Silverstein [43], Rider and Virág [42], Jiang [32], and Hachem et. al. =-=[28, 29]-=- provide several illuminating insights and new results. In this paper we introduce a result (Theorem 3.1) that may provide a unified ‘soft tool’ for matrices that can be easily expressed as smooth fun... |

13 | On the spectral norm of a random Toeplitz matrix
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- 2007
(Show Context)
Citation Context ...ssian random variables. Let An be the matrix n∑ An := n −1/2 (X |i−j|)1≤i,j≤n. This is a gaussian Toeplitz matrix, of the kind recently considered in Bryc, Dembo, and Jiang [13] and also in M. Meckes =-=[40]-=- and Bose and Sen [11]. n . □FLUCTUATIONS OF EIGENVALUES 17 Although Toeplitz determinants have been extensively studied (see e.g. Basor [7] and references therein), to the best of our knowledge, the... |

11 |
On an inequality and a characterization of the normal distribution connected with it
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- 1983
(Show Context)
Citation Context ...ifficult to construct zerobias transforms (not known at present for linear statistics of eigenvalues), Lemma 1.1 gives a direct formula for h. The second related idea is the work of Borovkov and Utev =-=[10]-=- which says that if a random variable W with E(W) = 0 and E(W 2 ) = 1 satisfies a Poincaré inequality with Poincaré constant close to 1, then W is approximately standard gaussian (if the Poincaré cons... |

10 | A CLT for information-theoretic statistics of gram random matrices with a given variance profile
- Hachem, Loubaton, et al.
- 2008
(Show Context)
Citation Context ...hnikov [47], Israelson [31] and Wieand [51]. The recent works of Anderson and Zeitouni [2], Dumitriu and Edelman [22], Rider and Silverstein [43], Rider and Virág [42], Jiang [32], and Hachem et. al. =-=[28, 29]-=- provide several illuminating insights and new results. In this paper we introduce a result (Theorem 3.1) that may provide a unified ‘soft tool’ for matrices that can be easily expressed as smooth fun... |

10 | Approximation of Haar distributed matrices and limiting distributions of eigenvalues of Jacobi ensembles
- Jiang
(Show Context)
Citation Context ... Hughes et. al. [30], Soshnikov [47], Israelson [31] and Wieand [51]. The recent works of Anderson and Zeitouni [2], Dumitriu and Edelman [22], Rider and Silverstein [43], Rider and Virág [42], Jiang =-=[32]-=-, and Hachem et. al. [28, 29] provide several illuminating insights and new results. In this paper we introduce a result (Theorem 3.1) that may provide a unified ‘soft tool’ for matrices that can be e... |

9 | Spectral norm of random large dimensional noncentral Toeplitz and
- Bose, Sen
(Show Context)
Citation Context .... Let An be the matrix n∑ An := n −1/2 (X |i−j|)1≤i,j≤n. This is a gaussian Toeplitz matrix, of the kind recently considered in Bryc, Dembo, and Jiang [13] and also in M. Meckes [40] and Bose and Sen =-=[11]-=-. n . □FLUCTUATIONS OF EIGENVALUES 17 Although Toeplitz determinants have been extensively studied (see e.g. Basor [7] and references therein), to the best of our knowledge, there are no existing cen... |

9 |
On random matrices from classical compact groups
- Johansson
- 1997
(Show Context)
Citation Context ... by Anderson and Zeitouni [1]. Other influential ideas, sometimes at varying levels of rigor, come from the papers of Costin and Lebowitz [19], Boutet de Monvel, Pastur and Shcherbina [12], Johansson =-=[33]-=-, Keating and Snaith [37], Hughes et. al. [30], Soshnikov [47], Israelson [31] and Wieand [51]. The recent works of Anderson and Zeitouni [2], Dumitriu and Edelman [22], Rider and Silverstein [43], Ri... |

8 |
Toeplitz determinants, Fisher-Hartwig symbols, and random matrices
- Basor
- 2005
(Show Context)
Citation Context ...ered in Bryc, Dembo, and Jiang [13] and also in M. Meckes [40] and Bose and Sen [11]. n . □FLUCTUATIONS OF EIGENVALUES 17 Although Toeplitz determinants have been extensively studied (see e.g. Basor =-=[7]-=- and references therein), to the best of our knowledge, there are no existing central limit theorems for general linear statistics of eigenvalues of random Toeplitz matrices. We have the following res... |

8 |
Spectral measure of large random
- Bryc, Dembo, et al.
- 2006
(Show Context)
Citation Context ...be independent standard gaussian random variables. Let An be the matrix n∑ An := n −1/2 (X |i−j|)1≤i,j≤n. This is a gaussian Toeplitz matrix, of the kind recently considered in Bryc, Dembo, and Jiang =-=[13]-=- and also in M. Meckes [40] and Bose and Sen [11]. n . □FLUCTUATIONS OF EIGENVALUES 17 Although Toeplitz determinants have been extensively studied (see e.g. Basor [7] and references therein), to the... |

7 | Gaussian fluctuations for non-Hermitian random matrix ensembles
- Rider, Silverstein
- 1986
(Show Context)
Citation Context ...son [33], Keating and Snaith [37], Hughes et. al. [30], Soshnikov [47], Israelson [31] and Wieand [51]. The recent works of Anderson and Zeitouni [2], Dumitriu and Edelman [22], Rider and Silverstein =-=[43]-=-, Rider and Virág [42], Jiang [32], and Hachem et. al. [28, 29] provide several illuminating insights and new results. In this paper we introduce a result (Theorem 3.1) that may provide a unified ‘sof... |

6 |
The central limit theorem and Poincaré-type inequalities
- Chen
- 1988
(Show Context)
Citation Context ...isfies a Poincaré inequality with Poincaré constant close to 1, then W is approximately standard gaussian (if the Poincaré constant is exactly 1, the W is exactly standard gaussian). As shown by Chen =-=[17]-=-, this fact can be used to prove central limit theorems in ways that are closely related to Stein’s method. Although it seems plausible, we could not detect any apparent relationship between this conc... |

5 | The noise in the circular law and the Gaussian free field
- Rider, Virág
(Show Context)
Citation Context ...Snaith [37], Hughes et. al. [30], Soshnikov [47], Israelson [31] and Wieand [51]. The recent works of Anderson and Zeitouni [2], Dumitriu and Edelman [22], Rider and Silverstein [43], Rider and Virág =-=[42]-=-, Jiang [32], and Hachem et. al. [28, 29] provide several illuminating insights and new results. In this paper we introduce a result (Theorem 3.1) that may provide a unified ‘soft tool’ for matrices t... |

4 | Perturbed Hankel determinants - Basor, Chen - 2005 |

4 | Large deviations and upper bounds for non-commutative functionals of Gaussian large random matrices - Guionnet - 2002 |