scholarly journals THE SPECTRUM OF RANDOM KERNEL MATRICES: UNIVERSALITY RESULTS FOR ROUGH AND VARYING KERNELS

2013 ◽  
Vol 02 (03) ◽  
pp. 1350005 ◽  
Author(s):  
YEN DO ◽  
VAN VU
Keyword(s):  
Large Class ◽  
Random Matrices ◽  
Spectral Distribution ◽  
Inner Product ◽  
Limiting Spectral Distribution ◽  
Product Kernel ◽  
Gaussian Vectors ◽  
Spectral Distributions ◽  
As If

We consider random matrices whose entries are [Formula: see text] or f(‖Xi – Xj‖2) for iid vectors Xi ∈ ℝp with normalized distribution. Assuming that f is sufficiently smooth and the distribution of Xi's is sufficiently nice, El Karoui [The spectrum of Kernel random matrices, Ann. Statist.38(1) (2010) 1–50, MR 2589315 (2011a.62187)] showed that the spectral distributions of these matrices behave as if f is linear in the Marčhenko–Pastur limit. When Xi's are Gaussian vectors, variants of this phenomenon were recently proved for varying kernels, i.e. when f may depend on p, by Cheng–Singer [The spectrum of random inner-product Kernel matrices, preprint (2012), arXiv:1202.3155 [math.PR]]. Two results are shown in this paper: first, it is shown that for a large class of distributions the regularity assumptions on f in El Karoui's results can be reduced to minimal; and second, it is shown that the Gaussian assumptions in Cheng–Singer's result can be removed, answering a question posed in [The spectrum of random inner-product Kernel matrices, preprint (2012), arXiv:1202.3155 [math.PR]] about the universality of the limiting spectral distribution.

scholarly journals Spectral distributions of periodic random matrix ensembles

2019 ◽  
Vol 10 (01) ◽  
pp. 2150011
Author(s):  
Roger Van Peski
Keyword(s):  
Random Matrix ◽  
Spectral Distribution ◽  
Eigenvalue Distribution ◽  
Limiting Spectral Distribution ◽  
Gaussian Unitary Ensemble ◽  
Random Matrix Ensembles ◽  
Spectral Distributions ◽  
Matrix Ensembles ◽  
General Correspondence ◽  
Unitary Ensemble

Koloğlu, Kopp and Miller compute the limiting spectral distribution of a certain class of real random matrix ensembles, known as [Formula: see text]-block circulant ensembles, and discover that it is exactly equal to the eigenvalue distribution of an [Formula: see text] Gaussian unitary ensemble. We give a simpler proof that under very general conditions which subsume the cases studied by Koloğlu–Kopp–Miller, real-symmetric ensembles with periodic diagonals always have limiting spectral distribution equal to the eigenvalue distribution of a finite Hermitian ensemble with Gaussian entries which is a ‘complex version’ of a [Formula: see text] submatrix of the ensemble. We also prove an essentially algebraic relation between certain periodic finite Hermitian ensembles with Gaussian entries, and the previous result may be seen as an asymptotic version of this for real-symmetric ensembles. The proofs show that this general correspondence between periodic random matrix ensembles and finite complex Hermitian ensembles is elementary and combinatorial in nature.

scholarly journals Limiting spectral distribution of a class of Hankel type random matrices

2015 ◽  
Vol 04 (03) ◽  
pp. 1550010
Author(s):  
Anirban Basak ◽  
Arup Bose ◽  
Soumendu Sundar Mukherjee
Keyword(s):  
Random Matrices ◽  
Spectral Distribution ◽  
Rayleigh Distribution ◽  
Circulant Matrices ◽  
Limiting Spectral Distribution ◽  
Atomic Distribution

We consider an indexed class of real symmetric random matrices which generalize the symmetric Hankel and Reverse Circulant matrices. We show that the limiting spectral distribution of these matrices exists almost surely and the limit is continuous in the index. We also study other properties of the limit and, in particular, explicitly characterize it for a certain subclass of matrices as a mixture of the atomic distribution at zero and the symmetrized Rayleigh distribution.

Polynomial generalizations of the sample variance-covariance matrix when pn−1 → 0

2016 ◽  
Vol 05 (04) ◽  
pp. 1650014 ◽  
Author(s):  
Monika Bhattacharjee ◽  
Arup Bose
Keyword(s):  
Covariance Matrix ◽  
Random Matrices ◽  
Random Matrix ◽  
Spectral Distribution ◽  
Probability Space ◽  
Sample Variance ◽  
Limiting Spectral Distribution ◽  
Matrix Polynomials ◽  
State Formula ◽  
Variance Covariance Matrix

Let [Formula: see text] be random matrices, where [Formula: see text] are independently distributed. Suppose [Formula: see text], [Formula: see text] are non-random matrices of order [Formula: see text] and [Formula: see text] respectively. Suppose [Formula: see text], [Formula: see text] and [Formula: see text]. Consider all [Formula: see text] random matrix polynomials constructed from the above matrices of the form [Formula: see text] [Formula: see text] and the corresponding centering polynomials [Formula: see text] [Formula: see text]. We show that under appropriate conditions on the above matrices, the variables in the non-commutative ∗-probability space [Formula: see text] with state [Formula: see text] converge. We also show that the limiting spectral distribution of [Formula: see text] exists almost surely whenever [Formula: see text] and [Formula: see text] are self-adjoint. The limit can be expressed in terms of, semi-circular, circular and other families and, limits of [Formula: see text], [Formula: see text] and non-commutative limit of [Formula: see text]. Our results fully generalize the results already known for [Formula: see text].

scholarly journals Limiting spectral distribution of large dimensional random matrices of linear processes

2020 ◽  
Vol 17 (2) ◽  
Author(s):  
Zahira Khettab
Keyword(s):  
Spectral Density ◽  
Numerical Simulations ◽  
Random Matrices ◽  
Spectral Distribution ◽  
Symmetric Matrix ◽  
Density Estimator ◽  
Linear Processes ◽  
Limiting Spectral Distribution ◽  
Dependence Conditions ◽  
Empirical Density

The limiting spectral distribution (LSD) of large sample radom matrices is derived under dependence conditions. We consider the matrices \(X_{N}T_{N}X_{N}^{\prime}\) , where \(X_{N}\) is a matrix (\(N \times n(N)\)) where the column vectors are modeled as linear processes, and \(T_{N}\) is a real symmetric matrix whose LSD exists. Under some conditions we show that, the LSD of \(X_{N}T_{N}X_{N}^{\prime}\) exists almost surely, as \(N \rightarrow \infty\) and \(n(N)/N \rightarrow c > 0\). Numerical simulations are also provided with the intention to study the convergence of the empirical density estimator of the spectral density of \(X_{N}T_{N}X_{N}^{\prime}\).

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