## A note on universality of the distribution of the largest eigenvalues in certain sample covariance matrices (2002)

Venue: | J. Statist. Phys |

Citations: | 60 - 3 self |

### BibTeX

@ARTICLE{Soshnikov02anote,

author = {Alexander Soshnikov},

title = {A note on universality of the distribution of the largest eigenvalues in certain sample covariance matrices},

journal = {J. Statist. Phys},

year = {2002},

volume = {108},

pages = {1033--1056}

}

### Years of Citing Articles

### OpenURL

### Abstract

Recently Johansson (21) and Johnstone (16) proved that the distribution of the (properly rescaled) largest principal component of the complex (real) Wishart matrix X g X(X t X) converges to the Tracy–Widom law as n, p (the dimensions of X) tend to. in some ratio n/p Q c>0.We extend these results in two directions. First of all, we prove that the joint distribution of the first, second, third, etc. eigenvalues of a Wishart matrix converges (after a proper rescaling) to the Tracy–Widom distribution. Second of all, we explain how the combinatorial machinery developed for Wigner random matrices in refs. 27, 38, and 39 allows to extend the results by Johansson and Johnstone to the case of X with non-Gaussian entries, provided n − p=O(p 1/3). We also prove that l max [ (n 1/2 +p 1/2) 2 +O(p 1/2 log(p)) (a.e.) for general c>0. KEY WORDS: Sample covariance matrices; principal component; Tracy– Widom distribution.

### Citations

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Citation Context ...Remark 1. The archetypical examples of sample covariance matrices is a p variate Wishart distribution on n degrees of freedom with identity covariance. It corresponds to in the real case, and x ij ’ N=-=(0, 1)-=-, 1 [ i [ n, 1 [ j [ p, (1.8) Re x ij, Im x ij ’ N(0, 1), 1 [ i [ n, 1 [ j [ p, (1.9) in the complex case. It was proved in refs. 14, 17, and 19 that if (i) ((iŒ) in the real case) is satisfied, n/p Q... |

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Citation Context ... )(n −1/2 + p −1/2 ) 1/3 (1.13) converges in distribution to the Tracy-Widom law ( F1 in the real case, F2 in the complex case). Remark 2 Tracy-Widom distribution was discovered by Tracy and Widom in =-=[33]-=-, [34]. They found that the limiting distribution of the (properly rescaled) largest eigenvalue of a Gaussian symmetric (Gaussian Hermitian) matrix is given by F1(F2), where F1(x) = exp{− 1 2 F2(x) = ... |

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Citation Context ...002) A Note on Universality of the Distribution of the Largest Eigenvalues in Certain Sample Covariance Matrices Alexander Soshnikov 1 Received April 20, 2001; revised June 7, 2001 Recently Johansson =-=(21)-=- and Johnstone (16) proved that the distribution of the (properly rescaled) largest principal component of the complex (real) Wishart matrix X g X(X t X) converges to the Tracy–Widom law as n, p (the ... |

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Citation Context ...1.13)) converge to the Tracy-Widom law with β = 1 (2) . Similar result for Wigner random matrices was proven in [30]. For other results on universality in random matrices we refer the reader to [26], =-=[11]-=-, [7], [20], [8], [22]. While we expect the result of Theorem 2 to be true whenever n/p → γ > 0 , we do not know at this moment how to extend our technique to the case of general γ . In this paper we ... |

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Citation Context ...p)) (a.e.) for general γ > 0. 1 Introduction Sample covariance matrices were introduced by statisticians about seventy years ago ([25], [39]). There is a large literature on the subject (see e.g. [2]-=-=[6]-=-, [9], [12]-[18], [21], [23], [36]-[37]). We start with the real case. 11.1 Real Sample Covariance Matrices The ensemble consists of p-dimensional random matrices Ap = X t X (X t denotes a transpose ... |

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Citation Context ...1/2 + p −1/2 ) 1/3 (1.13) converges in distribution to the Tracy-Widom law ( F1 in the real case, F2 in the complex case). Remark 2 Tracy-Widom distribution was discovered by Tracy and Widom in [33], =-=[34]-=-. They found that the limiting distribution of the (properly rescaled) largest eigenvalue of a Gaussian symmetric (Gaussian Hermitian) matrix is given by F1(F2), where F1(x) = exp{− 1 2 F2(x) = exp{− ... |

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Citation Context ...cations to combinatorics, representation theory, probability, statistics, mathematical physics, in which Tracy-Widom law appears as a limiting distribution ( for recent surveys we refer the reader to =-=[1]-=-, [10]). Remark 3 It should be noted that Johansson studied the complex case and Johnstone did the real case. Johnstone also gave an alternative proof in the complex case. We also note that Johnstone ... |

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Citation Context ... converge to the Tracy-Widom law with β = 1 (2) . Similar result for Wigner random matrices was proven in [30]. For other results on universality in random matrices we refer the reader to [26], [11], =-=[7]-=-, [20], [8], [22]. While we expect the result of Theorem 2 to be true whenever n/p → γ > 0 , we do not know at this moment how to extend our technique to the case of general γ . In this paper we settl... |

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Citation Context ...oof the asymptotics of Laguerre polynomials. Remark 4 On a physical level of rigor the results similar to those from the Johansson-Johnstone Theorem (in the complex case) were derived by Forrester in =-=[13]-=-. While it was not specifically pointed there, the results obtained in [21] imply that the joint distribution of the first, second, third , . . . , k-th, k = 1, 2, . . . largest eigenvalues converges ... |

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Citation Context ...values of a Wishart matrix converges (after a proper rescaling) to the Tracy-Widom distribution. Second of all, we explain how the combinatorial machinery developed for Wigner random matrices in [28]-=-=[30]-=- allows to extend the results by Johansson and Johnstone to the case of X with non-Gaussian entries, provided n − p = O(p 1/3 ). We also prove that λmax ≤ (n 1/2 + p 1/2 ) 2 + O(p 1/2 log(p)) (a.e.) f... |

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Citation Context ...s assume that p [ n, however our results remain valid for p>nas well. The distribution of the largest eigenvalues attracts a special attention (see, e.g., ref. 16, Section 1.2). It was shown by Geman =-=(12)-=- in the i.i.d. case that if E |x ij| 6+d < . the largest eigenvalue of A p/n converges to (1+c −1/2 ) 2 almost surely. A few years later Yin, Bai, Krishnaiah and Silverstein (4, 18) showed (in the i.i... |

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Citation Context ... to write the k-point correlation functions as ρ (p) k (x1, . . . , xk) = det 1≤i,j≤k Sp(xi, xj), k = 1, 2, . . . , p (2.5) (for more information on correlation functions we refer the reader to [24], =-=[35]-=-, [36]). As a by-product of the results in [21] Johnstone showed that after the rescaling the (rescaled) kernel x = µn,p + σn,ps (2.6) σn,pSp(µn,p + σn,ps1, µn,p + σn,ps2) (2.7) 7converges to the Air... |

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Citation Context ...It was shown by Geman (12) in the i.i.d. case that if E |x ij| 6+d < . the largest eigenvalue of A p/n converges to (1+c −1/2 ) 2 almost surely. A few years later Yin, Bai, Krishnaiah and Silverstein =-=(4, 18)-=- showed (in the i.i.d. case) that the finiteness of the fourth moment is a necessary and sufficient condition for the almost sure convergence (see also ref. 20). These results state that l max(A p)=(n... |

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Citation Context ...OF THEOREM 3 In order to estimate the r.h.s. of (3.4) we assume some familiarity of the reader with the combinatorial machinery developed in refs. 27, 38, and 39. In particular we refer the reader to =-=[28]-=- (Section 2, Definitions 1 and 2) or ref. 39 (Section 4, Definitions 1–4) how we defined (a) marked and unmarked instants, (b) a partition of all verices into the classes N0, N1,..., Nm and c) paths o... |

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Citation Context ...n by the r.h.s. of i,j=1,2 (2.18)-(2.21). The limiting correlation functions coincide with the limiting correlation functions at the edge of the spectrum in the Gaussian Orthogonal Ensemble (see e.g. =-=[14]-=-) (it also should be noted that the formulas (1.15)- (1.16) we gave in [30] for K(s, t) must be replaced by (2.18)-(2.21)). Proof of Lemma 1 The proof is a consequence of (2.11)-(2.14), (1.12)-(1.13) ... |

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Citation Context ...erge to the Tracy-Widom law with β = 1 (2) . Similar result for Wigner random matrices was proven in [30]. For other results on universality in random matrices we refer the reader to [26], [11], [7], =-=[20]-=-, [8], [22]. While we expect the result of Theorem 2 to be true whenever n/p → γ > 0 , we do not know at this moment how to extend our technique to the case of general γ . In this paper we settle for ... |

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Citation Context ...ersality of the Distribution of the Largest Eigenvalues in Certain Sample Covariance Matrices Alexander Soshnikov 1 Received April 20, 2001; revised June 7, 2001 Recently Johansson (21) and Johnstone =-=(16)-=- proved that the distribution of the (properly rescaled) largest principal component of the complex (real) Wishart matrix X g X(X t X) converges to the Tracy–Widom law as n, p (the dimensions of X) te... |

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Citation Context ...now follows from Lemma 2, part d) and Corollary 3. 4 Proof of Theorem 3 In order to estimate the r.h.s. of (3.4) we assume some familiarity of the reader with the combinatorial machinery developed in =-=[28]-=--[30]. In particular we refer the reader to [28] ( Section 2, Definitions 1-2) or [29] ( Section 4, Definitions 1-4) how we defined a) marked and unmarked instants, b) a partition of all verices into ... |

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Citation Context ...satisfied the machinery from [28]-[30] does not work, essentially for the following reason: when we decide which vertice to choose during the moment of self-intersection (as explained in section 4 of =-=[29]-=-) the number of choices for odd moments of time is smaller because of the constrain C1. If we now use the same bound as for the even moments of time (the one similar to the bound at the bottom of p.72... |

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Citation Context ...and (2.21). (2.19) immediately follows from (2.14) and (2.18). (2.20) is established in a similar way to (2.18), (2.21). To prove (2.18) we employ a very useful integral representation for S p(x, y): =-=(37)-=- Sp(x, y)=F +. f(x+z) k(y+z)+k(x+z) f(y+z) dz, (2.22) 0 where f(x), k(x) are defined in (2.15)–(2.17). The asymptotic behavior of f(x), k(x) was studied by Johnstone (16) who proved sn, pf(mn, p+sn, p... |

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Citation Context ...o the Tracy-Widom law with β = 1 (2) . Similar result for Wigner random matrices was proven in [30]. For other results on universality in random matrices we refer the reader to [26], [11], [7], [20], =-=[8]-=-, [22]. While we expect the result of Theorem 2 to be true whenever n/p → γ > 0 , we do not know at this moment how to extend our technique to the case of general γ . In this paper we settle for a wea... |

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Citation Context ...ntity covariance. It corresponds to in the real case, and to in the complex case. xij ∼ N(0, 1), 1 ≤ i ≤ n, 1 ≤ j ≤ p, (1.8) Rexij, Imxij ∼ N(0, 1), 1 ≤ i ≤ n, 1 ≤ j ≤ p, (1.9) It was proved in [23], =-=[17]-=-, [37] that if (i) ((i’) in the real case) is satisfied, n/p → γ ≥ 1, as p → ∞, and E|xij| 2+δ < const (1.10) then the empirical distribution function of the eigenvalues of Ap/n converges to a non-ran... |

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Citation Context ...a.e.) for general γ > 0. 1 Introduction Sample covariance matrices were introduced by statisticians about seventy years ago ([25], [39]). There is a large literature on the subject (see e.g. [2]-[6], =-=[9]-=-, [12]-[18], [21], [23], [36]-[37]). We start with the real case. 11.1 Real Sample Covariance Matrices The ensemble consists of p-dimensional random matrices Ap = X t X (X t denotes a transpose matri... |

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Citation Context ...Yin, Bai, Krishnaiah and Silverstein ([36], [3]) showed (in the i.i.d. case) that the finiteness of the fourth moment is a necessary and sufficient condition for the almost sure convergence (see also =-=[27]-=-). These results state that λmax(Ap) = (n 1/2 +p 1/2 ) 2 +o(n+p). However no results were known about the rate of the convergence until recently Johansson ([19]) and Johnstone ([21]) proved the follow... |

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Citation Context ...y) (2.3) is the reproducing (Christoffel-Darboux) kernel of the Laguerre orthonormalized system ϕ (αp) √ j! j (x) = (j + αp)! xαp/2 exp(−x/2)L αp j (x), (2.4) and L αp j are the Laguerre polynomials (=-=[32]-=-). This allows one to write the k-point correlation functions as ρ (p) k (x1, . . . , xk) = det 1≤i,j≤k Sp(xi, xj), k = 1, 2, . . . , p (2.5) (for more information on correlation functions we refer th... |

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Citation Context ...log(p)) (a.e.) for general γ > 0. 1 Introduction Sample covariance matrices were introduced by statisticians about seventy years ago ([25], [39]). There is a large literature on the subject (see e.g. =-=[2]-=--[6], [9], [12]-[18], [21], [23], [36]-[37]). We start with the real case. 11.1 Real Sample Covariance Matrices The ensemble consists of p-dimensional random matrices Ap = X t X (X t denotes a transp... |

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Citation Context ... j (y) (2.3) is the reproducing (Christoffel–Darboux) kernel of the Laguerre orthonormalized system j (ap) j (x)== j! (j+ap)! xap/2 ap exp(−x/2) Lj (x), (2.4) and L ap j are the Laguerre polynomials. =-=(35)-=- This allows one to write the k-point correlation functions as r (p) k (x1,..., xk)= det Sp(xi,xj), k=1, 2,..., p (2.5) 1 [ i, j [ k (for more information on correlation functions we refer the reader ... |

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Citation Context ...es in Certain Sample Covariance Matrices Alexander Soshnikov University of California Department of Mathematics One Shields Avenue Davis, CA 95616 USA Abstract Recently Johansson [19]) and Johnstone (=-=[21]-=-) proved that the distribution of the (properly rescaled) largest principal component of the complex (real) Wishart matrix X ∗ X (X t X) converges to the TracyWidom law as n, p (the dimensions of X) t... |

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Citation Context ...istribution of the Largest Eigenvalues in Sample Covariance Matrices 1039 correlation functions, k=1, 2,... . In the complex case the density of the joint distribution of the eigenvalues is given by: =-=(15)-=- P p(x 1,..., x p)=c −1 n, p D 1 [ i<j[ p (xi −xj) 2 p D x j=1 ap j exp(−xj), ap=n − p, (2.1) where c n, p is a normalization constant. Using a standard argument from Random Matrix Theory (34) one can... |

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Citation Context ... Tracy-Widom law with β = 1 (2) . Similar result for Wigner random matrices was proven in [30]. For other results on universality in random matrices we refer the reader to [26], [11], [7], [20], [8], =-=[22]-=-. While we expect the result of Theorem 2 to be true whenever n/p → γ > 0 , we do not know at this moment how to extend our technique to the case of general γ . In this paper we settle for a weaker re... |

1 |
T.Nagao, G.Honner, Correlations for the orthogonalunitary and symplectic-unitary transitions at the hard and soft
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(Show Context)
Citation Context ...n by the r.h.s. of i,j=1,2 (2.18)-(2.21). The limiting correlation functions coincide with the limiting correlation functions at the edge of the spectrum in the Gaussian Orthogonal Ensemble (see e,g. =-=[14]-=-) (it also should be noted that the formulas (1.15)- (1.16) we gave in [30] for K(s, t) must be replaced by (2.18)-(2.21)). Proof of Lemma 1 The proof is a consequence of (2.11)-(2.14), (1.12)-(1.13) ... |