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.

Patterned sparse random matrices: A moment approach

2017 ◽  
Vol 06 (03) ◽  
pp. 1750011
Author(s):  
Debapratim Banerjee ◽  
Arup Bose
Keyword(s):  
Random Matrices ◽  
Limit Distribution ◽  
Spectral Distribution ◽  
Explicit Description ◽  
Circulant Matrices ◽  
Hankel Matrices ◽  
Largest Eigenvalue ◽  
Empirical Spectral Distribution ◽  
Moment Approach ◽  
The Largest Eigenvalue

We consider four specific [Formula: see text] sparse patterned random matrices, namely the Symmetric Circulant, Reverse Circulant, Toeplitz and the Hankel matrices. The entries are assumed to be Bernoulli with success probability [Formula: see text] such that [Formula: see text] with [Formula: see text]. We use the moment approach to show that the expected empirical spectral distribution (EESD) converges weakly for all these sparse matrices. Unlike the Sparse Wigner matrices, here the random empirical spectral distribution (ESD) converges weakly to a random distribution. This weak convergence is only in the distribution sense. We give explicit description of the random limits of the ESD for Reverse Circulant and Circulant matrices. As in the non-sparse case, explicit description of the limits appears to be difficult to obtain in the Toeplitz and Hankel cases. We provide some properties of these limits. We then study the behavior of the largest eigenvalue of these matrices. We prove that for the Reverse Circulant and Symmetric Circulant matrices the limit distribution of the largest eigenvalue is a multiple of the Poisson. For Toeplitz and Hankel matrices we show that the non-degenerate limit distribution exists, but again it does not seem to be easy to obtain any explicit description.

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 \(X_{N}T_{N}X_{N}^{\prime}\).

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.

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