## Concentration of the Spectral Measure for Large Random Matrices with Stable Entries (2007)

Citations: | 1 - 0 self |

### BibTeX

@MISC{Houdré07concentrationof,

author = {Christian Houdré and Hua Xu},

title = {Concentration of the Spectral Measure for Large Random Matrices with Stable Entries},

year = {2007}

}

### OpenURL

### Abstract

We derive concentration inequalities for functions of the empirical measure of large random matrices with infinitely divisible entries and, in particular, stable ones. We also give concentration results for some other functionals of these random matrices, such as the largest eigenvalue or the largest singular value.

### Citations

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Citation Context ...own explicit form. However the dependence structure among the entries of Y ∗ Y prevents the vector of entries to be, itself, infinitely divisible (this is a well known fact originating with Lévy, see =-=[25]-=-). The methodology we previously used cannot be directly applied to deal with functions of eigenvalues of Y ∗ Y. However, concentration results can be obtained when we consider the following facts, du... |

465 |
Real Analysis and Probability
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Citation Context ...that � sup �trN(g) − E g∈Lipb(1) N [trN(g)] � � δ ≥ 4 . Hence, P N � sup � �trN(f) − m(trN(f)) f∈Lipb(1) � � � ≥ δ ≤ P N � � sup �trN(g) − E g∈Lipb(1) N [trN(g)] � � � δ ≥ . 4 (1.8) Next, recall (see =-=[6]-=-, [17]) that the Wasserstein distance between any two probability measures µ1 and µ2 on R is defined by � � � � dW (µ1, µ2) = sup � fdµ1 − � � fdµ2�. (1.9) f∈Lipb(1) 8 R RsHence, Theorem 1.2 actually ... |

326 |
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Citation Context ...i . It is a classical result sometimes called Lidskii’s theorem ([24]), that the map MN×N(C) → S N which associates to each Hermitian matrix its ordered list of real eigenvalues is 1-Lipschitz ([10], =-=[17]-=-). For a matrix XA under consideration with eigenvalues λ(XA), it is then clear that the map ϕ : (ωR i,i, ωR i,j, ωI N 2 i,j)1≤i<j≤N ↦→ λ(XA) is Lipschitz, from (R , �·�) to (SN , �·�), with Lipschitz... |

265 |
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Citation Context ...in the simplex S N = {λ1 ≥ · · · ≥ λN : λi ∈ R, 1 ≤ i ≤ N}, where throughout S N is equipped with the Euclidian norm �λ� = � �N i=1 λ2 i . It is a classical result sometimes called Lidskii’s theorem (=-=[24]-=-), that the map MN×N(C) → S N which associates to each Hermitian matrix its ordered list of real eigenvalues is 1-Lipschitz ([10], [17]). For a matrix XA under consideration with eigenvalues λ(XA), it... |

242 | Widom H.: Level-Spacing Distributions and the Airy Kernel
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Citation Context ...re@math.gatech.edu † Georgia Institute of Technology, School of Mathematics, Atlanta, Georgia, 30332-0160, xu@math.gatech.edu 1sentries satisfying some moment conditions (Wigner [32], Tracy and Widom =-=[30]-=-, Soshnikov [26], Girko [7], Pastur [23], Bai [2], Götze and Tikhomirov [8]). There is relatively little work outside the independent or finite second moment assumptions. Let us mention Soshnikov [28]... |

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Citation Context ... Statements of Results: Large random matrices have recently attracted a lot of attention in fields such as statistics, mathematical physics or combinatorics (e.g., see Mehta [22], Bai and Silverstein =-=[3]-=-, Johnstone [16], Anderson, Guionnet and Zeitouni [1]). For various classes of matrix ensembles, the asymptotic behavior of the, properly centered and normalized, spectral measure or of the largest ei... |

178 |
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Citation Context ...are iid centered random variables with finite variance σ 2 , the empirical distribution of the eigenvalues of Y ∗ Y/N converges as K → ∞, N → ∞, and K/N → γ ∈ (0, +∞) to the Marčenko-Pastur law ([3], =-=[21]-=-) with density pγ(x) = 1 2πxγσ2 � (c2 − x)(x − c1), c1 ≤ x ≤ c2, where c1 = σ 2 (1 − γ −1/2 ) 2 and c2 = σ 2 (1 + γ −1/2 ) 2 . When the entries of Y are iid Gaussian, Johansson [14] and Johnstone [15]... |

172 |
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Citation Context ...rgia, 30332-0160, houdre@math.gatech.edu † Georgia Institute of Technology, School of Mathematics, Atlanta, Georgia, 30332-0160, xu@math.gatech.edu 1sentries satisfying some moment conditions (Wigner =-=[32]-=-, Tracy and Widom [30], Soshnikov [26], Girko [7], Pastur [23], Bai [2], Götze and Tikhomirov [8]). There is relatively little work outside the independent or finite second moment assumptions. Let us ... |

139 |
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Citation Context ...ration. 1 Introduction and Statements of Results: Large random matrices have recently attracted a lot of attention in fields such as statistics, mathematical physics or combinatorics (e.g., see Mehta =-=[22]-=-, Bai and Silverstein [3], Johnstone [16], Anderson, Guionnet and Zeitouni [1]). For various classes of matrix ensembles, the asymptotic behavior of the, properly centered and normalized, spectral mea... |

111 |
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Citation Context ...o-Pastur law ([3], [21]) with density pγ(x) = 1 2πxγσ2 � (c2 − x)(x − c1), c1 ≤ x ≤ c2, where c1 = σ 2 (1 − γ −1/2 ) 2 and c2 = σ 2 (1 + γ −1/2 ) 2 . When the entries of Y are iid Gaussian, Johansson =-=[14]-=- and Johnstone [15] showed, in the complex and real case respectively, that the properly normalized largest eigenvalue converges in distribution to the Tracy-Widom law ([30], [31]). Soshnikov [27] ext... |

101 | Universality at the edge of the spectrum in Wigner random matrices
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(Show Context)
Citation Context ...du † Georgia Institute of Technology, School of Mathematics, Atlanta, Georgia, 30332-0160, xu@math.gatech.edu 1sentries satisfying some moment conditions (Wigner [32], Tracy and Widom [30], Soshnikov =-=[26]-=-, Girko [7], Pastur [23], Bai [2], Götze and Tikhomirov [8]). There is relatively little work outside the independent or finite second moment assumptions. Let us mention Soshnikov [28] who, using the ... |

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(Show Context)
Citation Context ...ular value of K × N rectangular random matrices with independent Cauchy entries, showing that the largest singular value of such a matrix is of order K 2 N 2 . On another front, Guionnet and Zeitouni =-=[9]-=-, gave concentration results for functionals of the empirical spectral measure for random matrices whose entries are independent and either satisfy a Logarithmic Sobolev inequality or are compactly su... |

62 | On the concentration of eigenvalues of random symmetric matrices
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Citation Context ...when it deviates from its mean. They also noted that their technique could be applied to prove results for the largest eigenvalue or for the spectral radius of such matrices. Alon, Krivelevich and Vu =-=[1]-=- further obtained concentration results for any of the eigenvalues of a Wigner matrix with uniformly bounded entries (see, Ledoux [20] for more developments and references). Our purpose in the present... |

60 | A note on universality of the distribution of the largest eigenvalues in certain sample covariance matrices
- Soshnikov
(Show Context)
Citation Context ...son [14] and Johnstone [15] showed, in the complex and real case respectively, that the properly normalized largest eigenvalue converges in distribution to the Tracy-Widom law ([30], [31]). Soshnikov =-=[27]-=- extended the result of Johnstone to Wishart matrix with Non-Gaussian entries under the condition that K − N = O(N 1/3 ) and that the moments of the entries do not grow too fast. Soshnikov and Fyodoro... |

48 | On the distribution of the largest principal component
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- 2000
(Show Context)
Citation Context ...[21]) with density pγ(x) = 1 2πxγσ2 � (c2 − x)(x − c1), c1 ≤ x ≤ c2, where c1 = σ 2 (1 − γ −1/2 ) 2 and c2 = σ 2 (1 + γ −1/2 ) 2 . When the entries of Y are iid Gaussian, Johansson [14] and Johnstone =-=[15]-=- showed, in the complex and real case respectively, that the properly normalized largest eigenvalue converges in distribution to the Tracy-Widom law ([30], [31]). Soshnikov [27] extended the result of... |

40 |
The spectrum of random matrices
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Citation Context ...f Technology, School of Mathematics, Atlanta, Georgia, 30332-0160, xu@math.gatech.edu 1sentries satisfying some moment conditions (Wigner [32], Tracy and Widom [30], Soshnikov [26], Girko [7], Pastur =-=[23]-=-, Bai [2], Götze and Tikhomirov [8]). There is relatively little work outside the independent or finite second moment assumptions. Let us mention Soshnikov [28] who, using the method of determinants, ... |

26 |
Remarks on deviation inequalities for functions of infinitely divisible random vectors
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Citation Context ...are not always available when further knowledge on the Lévy measure of X is lacking. 2 Proofs: We start with a proposition, which is a direct consequence of the concentration inequalities obtained in =-=[11]-=- for general Lipschitz function of infinitely divisible random vectors with finite exponential moment. Proposition 2.1 Let X = (ωR i,i , ωR i,j , ωI i,j )1≤i<j≤N be a random vector with joint law PN ∼... |

25 | High dimensional statistical inference and random matrices
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Citation Context ...Results: Large random matrices have recently attracted a lot of attention in fields such as statistics, mathematical physics or combinatorics (e.g., see Mehta [22], Bai and Silverstein [3], Johnstone =-=[16]-=-, Anderson, Guionnet and Zeitouni [1]). For various classes of matrix ensembles, the asymptotic behavior of the, properly centered and normalized, spectral measure or of the largest eigenvalue is unde... |

14 | Poisson statistics for the largest eigenvalues of Wigner random matrices with heavy tails
- Soshnikov
- 2004
(Show Context)
Citation Context ...[30], Soshnikov [26], Girko [7], Pastur [23], Bai [2], Götze and Tikhomirov [8]). There is relatively little work outside the independent or finite second moment assumptions. Let us mention Soshnikov =-=[28]-=- who, using the method of determinants, studied the distribution of the largest eigenvalue of Wigner matrices with entries having heavy tails. (Recall that a real (or complex) Wigner matrix is a symme... |

11 | On the concentration of measure phenomenon for stable and related random vectors
- Houdré, Marchal
- 2002
(Show Context)
Citation Context ...g factor such that lim N→∞ N 2 L(bN)/b α N λmax(b −1 N M) is the largest eigenvalue of b−1 N M. In fact lim N→∞ = 2 and where N 2 α −ɛ /bN = 0 and 2 lim bN/N α N→∞ +ɛ = 0, for any ɛ > 0. As stated in =-=[12]-=-, when the random vector 11sX is in the domain of attraction of an α-stable distribution, concentration inequalities similar to (1.14) or (1.15) can be obtained for general Lipschitz function. In part... |

10 |
Spectral theory of random matrices
- Girko
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(Show Context)
Citation Context ... Institute of Technology, School of Mathematics, Atlanta, Georgia, 30332-0160, xu@math.gatech.edu 1sentries satisfying some moment conditions (Wigner [32], Tracy and Widom [30], Soshnikov [26], Girko =-=[7]-=-, Pastur [23], Bai [2], Götze and Tikhomirov [8]). There is relatively little work outside the independent or finite second moment assumptions. Let us mention Soshnikov [28] who, using the method of d... |

10 |
Deviation inequalities on largest eigenvalues, Geometric aspects of functional analysis
- Ledoux
- 2007
(Show Context)
Citation Context ...for the spectral radius of such matrices. Alon, Krivelevich and Vu [1] further obtained concentration results for any of the eigenvalues of a Wigner matrix with uniformly bounded entries (see, Ledoux =-=[20]-=- for more developments and references). Our purpose in the present work is to deal with matrices whose entries form a general infinitely divisible vector, and in particular a stable one (without indep... |

8 | On the largest singular values of random matrices with independent Cauchy entries
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(Show Context)
Citation Context ...x with entries belonging to the domain of attraction of an α-stable law, limN→∞ P N (λmax ≤ x) = exp (−x −α ) (here λmax is the largest eigenvalue of such a normalized matrix). Soshnikov and Fyodorov =-=[29]-=- further derived results for the largest singular value of K × N rectangular random matrices with independent Cauchy entries, showing that the largest singular value of such a matrix is of order K 2 N... |

7 |
Dimension free and infinite variance tail estimates on Poisson space. 2004. Available at ArXiv
- Breton, Houdré, et al.
(Show Context)
Citation Context ...ext. In particular, if the entries of the matrix Y are iid α-stable random variables, the largest eigenvalue of Y ∗ Y is of order K 2/α N 2/α . (ii) Let X ∼ ID(β, 0, ν) in Rd , then (see Lemma 5.4 in =-=[5]-=-) for any x > 0, and any norm � · �N on Rd , P � �X�N ≥ x � ≥ 1 � � 1 − exp − ν 4 �� u ∈ R d : �u�N ≥ 2x ���� . 19sBut, λmax(M1/2 ) is a norm of the vector X = (YR i,j , YI i,j ), which we denote by �... |

3 |
On finite range stable-type concentration
- Breton, Houdré
- 2007
(Show Context)
Citation Context ...−1 )C(α) α + eα)/α(2 − α). and where C(α) = 4 α (2 − It is also possible to obtain concentration results for smaller values of δ. The lower and intermediate range for the stable deviation obtained in =-=[4]-=- provide the appropriate tools to achieve the following result. We refer to [4] for complete arguments, and only provide below a sample result. Theorem 1.12 Let X = (ωR i,i, ωR i,j, ωI i,j)1≤i<j≤N be ... |

3 |
Deviation inequalities on largest eigenvalues. Lecture Notes in Math
- LEDOUX
- 2007
(Show Context)
Citation Context ...ithmic Sobolev inequality or are compactly supported. They obtained in that context, the subgaussian decay of the tails of the empirical spectral measure when deviating from its mean (see also Ledoux =-=[18]-=-). Our purpose in the present work is to deal with matrices whose entries form a general infinitely divisible vector, and in particular a stable one. We obtain concentration results for functionals of... |

3 |
On orthorgonal and symplectic random matrix ensembles
- Tracy, Widom
- 1996
(Show Context)
Citation Context ... Gaussian, Johansson [14] and Johnstone [15] showed, in the complex and real case respectively, that the properly normalized largest eigenvalue converges in distribution to the Tracy-Widom law ([30], =-=[31]-=-). Soshnikov [27] extended the result of Johnstone to Wishart matrix with Non-Gaussian entries under the condition that K − N = O(N 1/3 ) and that the moments of the entries do not grow too fast. Sosh... |

2 |
Measure concentration for stable laws with index close to 2
- Marchal
(Show Context)
Citation Context ...er is thus consistent with the one in (1.26), as α is close to 2. Taking into account part (ii) above, the order of the constants in (1.24) are correct when α → 2. Following [4] (see also Remark 4 in =-=[19]-=-), we can recover a suboptimal Gaussian result by considering a particular stable random vector X (α) and letting α → 2. Toward this end, let X (α) be the stable random vector whose Lévy measure has f... |

2 |
L 1 -norms of infinitely divisible random vectors and certain stochastic integrals
- Marcus, Rosiński
(Show Context)
Citation Context ... �X� � ≤ N max j=1,2,...,N 2EN� X 2� j , (1.6) 7swhere the Xj, j = 1, 2, . . . , N 2 are the components of X. Actually, an estimate more precise than (1.6) is given by a result of Marcus and Rosiński =-=[20]-=- which asserts that if E[X] = 0, then 1 4 x0 ≤ E � �X� � ≤ 17 8 x0, where x0 is the solution of the equation: V 2 (x) x 2 M(x) + x where V 2 (x) is as before, while M(x) = � = 1, (1.7) �u�≥x �u�ν(du),... |

2 | concentration and fluctuations for Lévy processes
- Houdré, Marchal, et al.
- 2007
(Show Context)
Citation Context ... to be 2−α 10 ( ) α α−1 α−1 α ( σ(S N2 −1 ) ) 1 α−1 ( ) α N √2a α−1 1 ( ) α . 72D∗ α−1 Next, as already mentioned, J2(α) can be replaced by E N [‖X‖]. In fact, according to (1.7) and an estimation in =-=[14]-=-, if E N [X] = 0, then 1 4(2 − α) 1/ασ(SN2 −1 ) 1/α ≤ E N [‖X‖] ≤ 17 8 ( (2 − α)(α − 1) ) 1/α σ(SN2 −1 ) 1/α . Finally, note that, as in the proof of Theorem 1.2 (ii), the second term ( in (2.54) is d... |

2 | Concentration for norms of infinitely divisible vectors with independent components
- Houdré, Marchal, et al.
- 2008
(Show Context)
Citation Context ...(iv) When the entries of X are independent, and under a finite exponential moment assumption, the dependency in N of the function h (above and below) can sometimes be improved. We refer the reader to =-=[15]-=- where some of these generic problems are discussed and tackled. Next, recall (see [7], [19]) that the Wasserstein distance between any two probability measures µ1 and µ2 on R is defined by ∫ ∫ ∣ dW(µ... |

1 |
Lecture notes on random matrices. SAMSI
- Anderson, Guionnet, et al.
- 2006
(Show Context)
Citation Context ...cently attracted a lot of attention in fields such as statistics, mathematical physics or combinatorics (e.g., see Mehta [22], Bai and Silverstein [3], Johnstone [16], Anderson, Guionnet and Zeitouni =-=[1]-=-). For various classes of matrix ensembles, the asymptotic behavior of the, properly centered and normalized, spectral measure or of the largest eigenvalue is understood. Many of these results hold tr... |

1 |
On the circular law. Preprint. Avalaible at Math arXiv 0702386
- Götze, Tikhomirov
- 2007
(Show Context)
Citation Context ... Atlanta, Georgia, 30332-0160, xu@math.gatech.edu 1sentries satisfying some moment conditions (Wigner [32], Tracy and Widom [30], Soshnikov [26], Girko [7], Pastur [23], Bai [2], Götze and Tikhomirov =-=[8]-=-). There is relatively little work outside the independent or finite second moment assumptions. Let us mention Soshnikov [28] who, using the method of determinants, studied the distribution of the lar... |

1 |
mean and concentration inequalities for Lévy processes. Preprint. Available at Math arXiv 0607022
- Houdré, Marchal, et al.
(Show Context)
Citation Context ...� 1 α−1 � � α N α−1 1 √2a � � α . 72D∗ α−1 We remind the reader that, as already mentioned, J2(α) can be replaced by E N [�X�]. According to the result of Marcus and Rosiński [20] and the estimate in =-=[13]-=-, if E N [X] = 0, then 1 4(2 − α) 1/α σ(SN 2 −1 ) 1/α ≤ E N [�X�] ≤ 17 8 � (2 − α)(α − 1) � 1/α σ(SN 2 −1 ) 1/α . Finally, note that, as in the proof of Theorem 1.2 (ii), the second term in (2.54) is ... |

1 |
Spectral analysis of large dimensional random matrices
- Probab
- 1997
(Show Context)
Citation Context ...gy, School of Mathematics, Atlanta, Georgia, 30332-0160, xu@math.gatech.edu 1sentries satisfying some moment conditions (Wigner [32], Tracy and Widom [30], Soshnikov [26], Girko [7], Pastur [23], Bai =-=[2]-=-, G"otze and Tikhomirov [8]). There is relatively little work outside the independent or finite second moment assumptions. Let us mention Soshnikov [28] who, using the method of determinants, studied ... |

1 |
L1-norms of infinitely divisible random vectors and certain stochastic integrals
- Marcus, Rosi'nski
(Show Context)
Citation Context ...ax j=1,2,...,N2E N \Theta X2j \Lambda , (1.6) 7swhere the Xj, j = 1, 2, . . . , N 2 are the components of X. Actually, an estimate more precise than (1.6) is given by a result of Marcus and Rosi'nski =-=[20]-=- which asserts that if E[X] = 0, then 1 4 x0 <= E\Theta kXk\Lambdas<= 17 8 x0, where x0 is the solution of the equation: V 2(x) x2 + M (x) x = 1, (1.7) where V 2(x) is as before, while M (x) = Rkuk>=x... |

1 |
Lecture notes on random matrices
- Anderson, Guionnet, et al.
- 1997
(Show Context)
Citation Context ...cently attracted a lot of attention in fields such as statistics, mathematical physics or combinatorics (e.g., see Mehta [22], Bai and Silverstein [3], Johnstone [16], Anderson, Guionnet and Zeitouni =-=[1]-=-). For various classes of matrix ensembles, the asymptotic behavior of the, properly centered and normalized, spectral measure or of the largest eigenvalue is understood. Many of these results hold tr... |