It has been recently shown that if $X$ is an $n\times N$ matrix whose entries are i.i.d. standard complex Gaussian and $l_1$ is the largest eigenvalue of $X^*X$, there exist sequences $m_{n,N}$ and $s_{n,N}$ such that $(l_1-m_{n,N})/s_{n,N}$ converges in distribution to $W_2$, the Tracy--Widom law appearing in the study of the Gaussian unitary ensemble. This probability law has a density which is known and computable. The cumulative distribution function of $W_2$ is denoted $F_2$. In this paper we show that, under the assumption that $n/N\to γ\in(0,\infty)$, we can find a function $M$, continuous and nonincreasing, and sequences $\tildeμ_{n,N}$ and $\tildeσ_{n,N}$ such that, for all real $s_0$, there exists an integer $N(s_0,γ)$ for which, if $(n\wedge N)\geq N(s_0,γ)$, we have, with $l_{n,N}=(l_1-\tildeμ_{n,N})/\tildeσ_{n,N}$, \[\forall s\geq s_0\qquad (n\wedge N)^{2/3}|P(l_{n,N}\leq s)-F_2(s)|\leq M(s_0)\exp(-s).\] The surprisingly good 2/3 rate and qualitative properties of the bounding function help explain the fact that the limiting distribution $W_2$ is a good approximation to the empirical distribution of $l_{n,N}$ in simulations, an important fact from the point of view of (e.g., statistical) applications.
Using a change-of-measure argument, we prove an equality in law between the process of largest eigenvalues in a generalized Wishart random-matrix process and a last-passage percolation process. This equality in law was conjectured by Borodin and Peche.
In this paper we consider the extreme behavior of the extremal eigenvalues of white Wishart matrices, which plays an important role in multivariate analysis. In particular, we focus on the case when the dimension of the feature p is much larger than or comparable to the number of observations n, a common situation in modern data analysis. We provide asymptotic approximations and bounds for the tail probabilities of the extremal eigenvalues. Moreover, we construct efficient Monte Carlo simulation algorithms to compute the tail probabilities. Simulation results show that our method has the best …
This paper extends the work of El Karoui [Ann. Probab. 35 (2007) 663--714] which finds the Tracy--Widom limit for the largest eigenvalue of a nonsingular $p$-dimensional complex Wishart matrix $W_{\mathbb{C}}(Ω_p,n)$ to the case of several of the largest eigenvalues of the possibly singular $(n<p)$ matrix $W_{\mathbb{C}}(Ω_p,n).$ As a byproduct, we extend all results of Baik, Ben Arous and Peche [Ann. Probab. 33 (2005) 1643--1697] to the singular Wishart matrix case. We apply our findings to obtain a 95% confidence set for the number of common risk factors in excess stock returns.
We solve the largest sample eigenvalue distribution problem in the rank 1 spiked model of the quaternionic Wishart ensemble, which is the first case of a statistical generalization of the Laguerre symplectic ensemble (LSE) on the soft edge. We observe a phase change phenomenon similar to that in the complex case, and prove that the new distribution at the phase change point is the GOE Tracy--Widom distribution.
In this paper, we consider N-dimensional real Wishart matrices Y in the class $W_{\mathbb{R}}(Σ,M)$ in which all but one eigenvalues of $Σ$ is 1. Let the non-trivial eigenvalue of $Σ$ be $1+τ$, then as N, $M\rightarrow\infty$, with $M/N=γ^2$ finite and non-zero, the eigenvalue distribution of $Y$ will converge into the Marchenko-Pastur distribution inside a bulk region. When $τ$ increases from zero, one starts to see a stray eigenvalue of Y outside of the support of the Marchenko-Pastur density. As the this stray eigenvalue leaves the bulk region, a phase transition will occur in the largest e…
This is the first part of a paper that studies the phase transition in the asymptotic limit of the rank 1 real Wishart spiked model. In this paper, we consider $N$-dimensional real Wishart matrices $S$ in the class $W_{\mathbb{R}}\left(Σ,M\right)$ in which all but one eigenvalues of $Σ$ is $1$. Let the non-trivial eigenvalue of $Σ$ be $1+τ$, then as $N$, $M\rightarrow\infty$, with $N/M=γ^2$ finite and non-zero, the eigenvalue distribution of $S$ will converge into the Machenko-Pastur distribution inside a bulk region. As $τ$ increases from zero, one starts seeing stray eigenvalues of $S$ outsi…
The spiked model is an important special case of the Wishart ensemble, and a natural generalization of the white Wishart ensemble. Mathematically, it can be defined on three kinds of variables: the real, the complex and the quaternion. For practical application, we are interested in the limiting distribution of the largest sample eigenvalue. We first give a new proof of the result of Baik, Ben Arous and Péché for the complex spiked model, based on the method of multiple orthogonal polynomials by Bleher and Kuijlaars. Then in the same spirit we present a new result of the rank 1 quaternionic sp…
Extreme principal minors of Wishart and deformed GOE matrices
School of Data Science, The Chinese University of Hong Kong, Shenzhen（香港中文大学深圳数据科学学院）；Department of Statistics and Data Science, University of Texas, Austin（德克萨斯大学奥斯汀分校统计与数据科学系）
We derive Painlevé--type expressions for the distribution of the $m^{th}$ largest eigenvalue in the Gaussian Orthogonal and Symplectic Ensembles in the edge scaling limit. This work generalizes to general $m$ the $m=1$ results of Tracy and Widom [23]. The results of Johnstone and Soshnikov (see [15], [19]) imply the immediate relevance of our formulas for the $m^{th}$ largest eigenvalue of the appropriate Wishart distribution.
We studied universality of Wishart ensembles whose covariance matrix has 2 distinct eigenvalues and the number of each of these eigenvalue goes to infinity in the asymptotic limit. In this case, the limiting eigenvalue distribution can be supported on 1 or 2 disjoint intervals. In our previous work the case when the support consists of 2 intervals was studied. This paper complements our previous analysis and studied the case when the support consists of a single interval. By using Riemann-Hilbert analysis, we have shown that under proper rescaling of the eigenvalues, the limiting correlation k…
We studied the universality of Wishart ensembles whose covariance matrix has 2 distinct eigenvalues. We studied the asymptotic limit when the number of both eigenvalues goes to infinity and obtained universality results. In this case, the limiting eigenvalue distribution can be supported on 1 or 2 disjoint intervals. We obtained a necessary and sufficient condition on the parameters such that the limiting distribution is supported on 2 disjoint intervals and have computed the eigenvalue density in the limit. Furthermore, by using Riemann-Hilbert analysis, we have shown that under proper rescal…
We establish a quantitative version of the Tracy--Widom law for the largest eigenvalue of high dimensional sample covariance matrices. To be precise, we show that the fluctuations of the largest eigenvalue of a sample covariance matrix $X^*X$ converge to its Tracy--Widom limit at a rate nearly $N^{-1/3}$, where $X$ is an $M \times N$ random matrix whose entries are independent real or complex random variables, assuming that both $M$ and $N$ tend to infinity at a constant rate. This result improves the previous estimate $N^{-2/9}$ obtained by Wang [73]. Our proof relies on a Green function comp…
Let $A$ and $B$ be independent, central Wishart matrices in $p$ variables with common covariance and having $m$ and $n$ degrees of freedom, respectively. The distribution of the largest eigenvalue of $(A+B)^{-1}B$ has numerous applications in multivariate statistics, but is difficult to calculate exactly. Suppose that $m$ and $n$ grow in proportion to $p$. We show that after centering and scaling, the distribution is approximated to second-order, $O(p^{-2/3})$, by the Tracy--Widom law. The results are obtained for both complex and then real-valued data by using methods of random matrix theory …
Recently Johansson and Johnstone 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 Tracy-Widom law as $ n, p $ (the dimensions of $ X $) tend to $ \infty $ in some ratio $ n/p \to γ>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 …
We apply the method of determinants to study the distribution of the largest singular values of large $ m \times n $ real rectangular random matrices with independent Cauchy entries. We show that statistical properties of the (rescaled by a factor of $ \frac{1}{m^2\*n^2}$)largest singular values agree in the limit with the statistics of the inhomogeneous Poisson random point process with the intensity $ \frac{1}π x^{-3/2} $ and, therefore, are different from the Tracy-Widom law. Among other corollaries of our method we show an interesting connection between the mathematical expectations of the…
We derive Painlevé--type expressions for the distribution of the $m^{th}$ largest eigenvalue in the Gaussian Orthogonal and Symplectic Ensembles in the edge scaling limit. The work of Johnstone and Soshnikov (see [7], [10]) implies the immediate relevance of our formulas for the $m^{th}$ largest eigenvalue of the appropriate Wishart distribution.
We consider matrices formed by a random $N\times N$ matrix drawn from the Gaussian Orthogonal Ensemble (or Gaussian Unitary Ensemble) plus a rank-one perturbation of strength $θ$, and focus on the largest eigenvalue, $x$, and the component, $u$, of the corresponding eigenvector in the direction associated to the rank-one perturbation. We obtain the large deviation principle governing the atypical joint fluctuations of $x$ and $u$. Interestingly, for $θ>1$, in large deviations characterized by a small value of $u$, i.e. $u<1-1/θ$, the second-largest eigenvalue pops out from the Wigner semi-circ…
Consider a standard white Wishart matrix with parameters $n$ and $p$. Motivated by applications in high-dimensional statistics and signal processing, we perform asymptotic analysis on the maxima and minima of the eigenvalues of all the $m \times m$ principal minors, under the asymptotic regime that $n,p,m$ go to infinity. Asymptotic results concerning extreme eigenvalues of principal minors of real Wigner matrices are also obtained. In addition, we discuss an application of the theoretical results to the construction of compressed sensing matrices, which provides insights to compressed sensing…
In this article, we consider random Wigner matrices, that is symmetric matrices such that the subdiagonal entries of Xn are independent, centered, and with variance one except on the diagonal where the entries have variance two. We prove that, under some suitable hypotheses on the laws of the entries, the law of the largest eigenvalue satisfies a large deviation principle with the same rate function as in the Gaussian case. The crucial assumption is that the Laplace transform of the entries must be bounded above by the Laplace transform of a centered Gaussian variable with same variance. This …
We show in this note that the asymptotic spectral distribution, location and distribution of the largest eigenvalue of a large class of random density matrices coincide with that of Wishart-type random matrices using proper scaling. As an application, we show that the asymptotic entropy production rate is logarithmic. These results are generalizations of those of Nechita, and Sommers and \. Zyczkowski.
Let $\mathbf{W}\in\mathbb{C}^{n\times n}$ be a {\it single-spiked} Wishart matrix in the class $\mathbf{W}\sim \mathcal{CW}_n(m,\mathbf{I}_n+ θ\mathbf{v}\mathbf{v}^\dagger) $ with $m\geq n$, where $\mathbf{I}_n$ is the $n\times n$ identity matrix, $\mathbf{v}\in\mathbb{C}^{n\times 1}$ is an arbitrary vector with unit Euclidean norm, $θ\geq 0$ is a non-random parameter, and $(\cdot)^\dagger$ represents the conjugate-transpose operator. Let $\mathbf{u}_1$ and $\mathbf{u}_n$ denote the eigenvectors corresponding to the samllest and the largest eigenvalues of $\mathbf{W}$, respectively. This paper…
We investigate the random eigenvalues coming from the beta-Laguerre ensemble with parameter p, which is a generalization of the real, complex and quaternion Wishart matrices of parameter (n,p). In the case that the sample size n is much smaller than the dimension of the population distribution p, a common situation in modern data, we approximate the beta-Laguerre ensemble by a beta-Hermite ensemble which is a generalization of the real, complex and quaternion Wigner matrices. As corollaries, when n is much smaller than p, we show that the largest and smallest eigenvalues of the complex Wishart…
This paper is the second chapter of three of the author's undergraduate thesis. In this paper, we consider the random matrix ensemble given by $(d_b, d_w)$-regular graphs on $M$ black vertices and $N$ white vertices, where $d_b \in [N^γ, N^{2/3 - γ}]$ for any $γ> 0$. We simultaneously prove that the bulk eigenvalue correlation statistics for both normalized adjacency matrices and their corresponding covariance matrices are stable for short times. Combined with an ergodicity analysis of the Dyson Brownian motion in another paper, this proves universality of bulk eigenvalue correlation statistic…
We consider the problem of detecting signals in the rank-one signal-plus-noise data matrix models that generalize the spiked Wishart matrices. We show that the principal component analysis can be improved by pre-transforming the matrix entries if the noise is non-Gaussian. As an intermediate step, we prove a sharp phase transition of the largest eigenvalues of spiked rectangular matrices, which extends the Baik-Ben Arous-Péché (BBP) transition. We also propose a hypothesis test to detect the presence of signal with low computational complexity, based on the linear spectral statistics, which mi…
We consider the Laguerre Unitary Ensemble (aka, Wishart Ensemble) of sample covariance matrices $A = XX^*$, where $X$ is an $N \times n$ matrix with iid standard complex normal entries. Under the scaling $n = N + \lfloor \sqrt{ 4 c N} \rfloor$, $c > 0$ and $N \rightarrow \infty$, we show that the rescaled fluctuations of the smallest eigenvalue, largest eigenvalue and condition number of the matrices $A$ are all given by the Tracy--Widom distribution ($β= 2$). This scaling is motivated by the study of the solution of the equation $Ax=b$ using the conjugate gradient algorithm, in the case that …
Given a large, high-dimensional sample from a spiked population, the top sample covariance eigenvalue is known to exhibit a phase transition. We show that the largest eigenvalues have asymptotic distributions near the phase transition in the rank-one spiked real Wishart setting and its general beta analogue, proving a conjecture of Baik, Ben Arous and Péché (2005). We also treat shifted mean Gaussian orthogonal and beta ensembles. Such results are entirely new in the real case; in the complex case we strengthen existing results by providing optimal scaling assumptions. One obtains the known li…
Consider a deterministic self-adjoint matrix X_n with spectral measure converging to a compactly supported probability measure, the largest and smallest eigenvalues converging to the edges of the limiting measure. We perturb this matrix by adding a random finite rank matrix with delocalized eigenvectors and study the extreme eigenvalues of the deformed model. We give necessary conditions on the deterministic matrix X_n so that the eigenvalues converging out of the bulk exhibit Gaussian fluctuations, whereas the eigenvalues sticking to the edges are very close to the eigenvalues of the non-pert…
We study global fluctuations for singular values of $M$-fold products of several right-unitarily invariant $N \times N$ random matrix ensembles. As $N \to \infty$, we show the fluctuations of their height functions converge to an explicit Gaussian field, which is log-correlated for $M$ fixed and has a white noise component for $M \to \infty$ jointly with $N$. Our technique centers on the study of the multivariate Bessel generating functions of these spectral measures, for which we prove a central limit theorem for global fluctuations via certain conditions on the generating functions. We apply…
Let X_N= (X_1^(N), ..., X_p^(N)) be a family of N-by-N independent, normalized random matrices from the Gaussian Unitary Ensemble. We state sufficient conditions on matrices Y_N =(Y_1^(N), ..., Y_q^(N)), possibly random but independent of X_N, for which the operator norm of P(X_N, Y_N, Y_N^*) converges almost surely for all polynomials P. Limits are described by operator norms of objects from free probability theory. Taking advantage of the choice of the matrices Y_N and of the polynomials P we get for a large class of matrices the "no eigenvalues outside a neighborhood of the limiting spectru…
We study sample covariance matrices of the form $W=\frac 1n C C^T$, where $C$ is a $k\times n$ matrix with i.i.d. mean zero entries. This is a generalization of so-called Wishart matrices, where the entries of $C$ are independent and identically distributed standard normal random variables. Such matrices arise in statistics as sample covariance matrices, and the high-dimensional case, when $k$ is large, arises in the analysis of DNA experiments. We investigate the large deviation properties of the largest and smallest eigenvalues of $W$ when either $k$ is fixed and $n\to \infty$, or $k_n\to \i…
