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Central limit theorems for linear spectral statistics of large dimensional F-matrices

Shurong Zheng (2012)

Annales de l'I.H.P. Probabilités et statistiques

In many applications, one needs to make statistical inference on the parameters defined by the limiting spectral distribution of an F matrix, the product of a sample covariance matrix from the independent variable array (Xjk)p×n1 and the inverse of another covariance matrix from the independent variable array (Yjk)p×n2. Here, the two variable arrays are assumed to either both real or both complex. It helps to find the asymptotic distribution of the relevant parameter estimators associated with the...

Central limit theorems for the brownian motion on large unitary groups

Florent Benaych-Georges (2011)

Bulletin de la Société Mathématique de France

In this paper, we are concerned with the large n limit of the distributions of linear combinations of the entries of a Brownian motion on the group of n × n unitary matrices. We prove that the process of such a linear combination converges to a Gaussian one. Various scales of time and various initial distributions are considered, giving rise to various limit processes, related to the geometric construction of the unitary Brownian motion. As an application, we propose a very short proof of the asymptotic...

Characteristic polynomials of sample covariance matrices: The non-square case

Holger Kösters (2010)

Open Mathematics

We consider the sample covariance matrices of large data matrices which have i.i.d. complex matrix entries and which are non-square in the sense that the difference between the number of rows and the number of columns tends to infinity. We show that the second-order correlation function of the characteristic polynomial of the sample covariance matrix is asymptotically given by the sine kernel in the bulk of the spectrum and by the Airy kernel at the edge of the spectrum. Similar results are given...

Chevet type inequality and norms of submatrices

Radosław Adamczak, Rafał Latała, Alexander E. Litvak, Alain Pajor, Nicole Tomczak-Jaegermann (2012)

Studia Mathematica

We prove a Chevet type inequality which gives an upper bound for the norm of an isotropic log-concave unconditional random matrix in terms of the expectation of the supremum of “symmetric exponential” processes, compared to the Gaussian ones in the Chevet inequality. This is used to give a sharp upper estimate for a quantity Γ k , m that controls uniformly the Euclidean operator norm of the submatrices with k rows and m columns of an isotropic log-concave unconditional random matrix. We apply these estimates...

Comparison between two types of large sample covariance matrices

Guangming Pan (2014)

Annales de l'I.H.P. Probabilités et statistiques

Let { X i j } , i , j = , be a double array of independent and identically distributed (i.i.d.) real random variables with E X 11 = μ , E | X 11 - μ | 2 = 1 and E | X 11 | 4 l t ; . Consider sample covariance matrices (with/without empirical centering) 𝒮 = 1 n j = 1 n ( 𝐬 j - 𝐬 ¯ ) ( 𝐬 j - 𝐬 ¯ ) T and 𝐒 = 1 n j = 1 n 𝐬 j 𝐬 j T , where 𝐬 ¯ = 1 n j = 1 n 𝐬 j and 𝐬 j = 𝐓 n 1 / 2 ( X 1 j , ... , X p j ) T with ( 𝐓 n 1 / 2 ) 2 = 𝐓 n , non-random symmetric non-negative definite matrix. It is proved that central limit theorems of eigenvalue statistics of 𝒮 and 𝐒 are different as n with p / n approaching a positive constant. Moreover, it is also proved that such a different behavior is not observed in the average behavior...

Computation of some examples of Brown's spectral measure in free probability

Philippe Biane, Franz Lehner (2001)

Colloquium Mathematicae

We use free probability techniques to compute spectra and Brown measures of some non-hermitian operators in finite von Neumann algebras. Examples include u + u where uₙ and u are the generators of ℤₙ and ℤ respectively, in the free product ℤₙ*ℤ, or elliptic elements of the form S α + i S β where S α and S β are free semicircular elements of variance α and β.

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