scholarly journals On optimal bounds in the local semicircle law under four moment condition

2019 ◽  
Vol 484 (3) ◽  
pp. 265-268
Author(s):  
F. Götze ◽  
A. A. Naumov ◽  
A. N. Tikhomirov
Keyword(s):  
Spectral Distribution ◽  
Moment Condition ◽  
Finite Moment ◽  
Semicircle Law ◽  
Empirical Spectral Distribution ◽  
Local Semicircle Law ◽  
Unit Variance ◽  
Optimal Bounds ◽  
Order Four ◽  
Stieltjes Transforms

We consider symmetric random matrices with independent mean zero and unit variance entries in the upper triangular part. Assuming that the distributions of matrix entries have finite moment of order four, we prove optimal bounds for the distance between the Stieltjes transforms of the empirical spectral distribution function and the semicircle law. Application concerning the convergence rate in probability of the empirical spectral distribution to the semicircle law is discussed as well.

On Optimal Bounds in the Local Semicircle Law under Four Moment Condition

2019 ◽  
Vol 99 (1) ◽  
pp. 40-43
Author(s):  
F. Götze ◽  
A. A. Naumov ◽  
A. N. Tikhomirov
Keyword(s):  
Moment Condition ◽  
Semicircle Law ◽  
Local Semicircle Law ◽  
Optimal Bounds

scholarly journals Local Semicircle Law Under Fourth Moment Condition

2019 ◽  
Vol 33 (3) ◽  
pp. 1327-1362
Author(s):  
F. Götze ◽  
A. Naumov ◽  
A. Tikhomirov
Keyword(s):  
Moment Condition ◽  
Fourth Moment ◽  
Semicircle Law ◽  
Local Semicircle Law

scholarly journals The circular law for random regular digraphs with random edge weights

2017 ◽  
Vol 06 (03) ◽  
pp. 1750012 ◽  
Author(s):  
Nicholas Cook
Keyword(s):  
Lebesgue Measure ◽  
Unit Disk ◽  
Directed Graph ◽  
Adjacency Matrix ◽  
Spectral Distribution ◽  
Hadamard Product ◽  
Empirical Spectral Distribution ◽  
Circular Law ◽  
Unit Variance ◽  
The Matrix

We consider random [Formula: see text] matrices of the form [Formula: see text], where [Formula: see text] is the adjacency matrix of a uniform random [Formula: see text]-regular directed graph on [Formula: see text] vertices, with [Formula: see text] for some fixed [Formula: see text], and [Formula: see text] is an [Formula: see text] matrix of i.i.d. centered random variables with unit variance and finite [Formula: see text]th moment (here ∘ denotes the matrix Hadamard product). We show that as [Formula: see text], the empirical spectral distribution of [Formula: see text] converges weakly in probability to the normalized Lebesgue measure on the unit disk.

Strong convergence of ESD for large quaternion sample covariance matrices and correlation matrices when p/n → 0

2019 ◽  
Vol 09 (02) ◽  
pp. 2050005
Author(s):  
Xue Ding
Keyword(s):  
Strong Convergence ◽  
Sample Size ◽  
Spectral Distribution ◽  
Covariance Matrices ◽  
Correlation Matrices ◽  
Semicircle Law ◽  
Empirical Spectral Distribution ◽  
Sample Covariance Matrices ◽  
Dimension Formula ◽  
Sample Covariance

In this paper, we study the strong convergence of empirical spectral distribution (ESD) of the large quaternion sample covariance matrices and correlation matrices when the ratio of the population dimension [Formula: see text] to sample size [Formula: see text] tends to zero. We prove that the ESD of renormalized quaternion sample covariance matrices converges almost surely to the semicircle law.

Local semicircle law under weak moment conditions

2016 ◽  
Vol 93 (3) ◽  
pp. 248-250 ◽  
Author(s):  
F. Götze ◽  
A. A. Naumov ◽  
A. N. Tikhomirov ◽  
D. A. Timushev
Keyword(s):  
Moment Conditions ◽  
Semicircle Law ◽  
Local Semicircle Law

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.

scholarly journals Local Semicircle Law in the Bulk for Gaussian β-Ensemble

2012 ◽  
Vol 148 (2) ◽  
pp. 204-232 ◽  
Author(s):  
Philippe Sosoe ◽  
Percy Wong
Keyword(s):  
Semicircle Law ◽  
Local Semicircle Law

On the limit of the spectral distribution of large-dimensional random quaternion covariance matrices

2017 ◽  
Vol 06 (02) ◽  
pp. 1750004 ◽  
Author(s):  
Yanqing Yin ◽  
Jiang Hu
Keyword(s):  
Machine Learning ◽  
Image Processing ◽  
Adaptive Filtering ◽  
Random Matrix Theory ◽  
Random Matrix ◽  
Spectral Distribution ◽  
Matrix Theory ◽  
Covariance Matrices ◽  
Moment Condition ◽  
Quaternion Matrices

The use of quaternions and quaternion matrices in practice, such as in machine learning, adaptive filtering, vector sensing and image processing, has recently been rapidly gaining in popularity. In this paper, by applying random matrix theory, we investigate the spectral distribution of large-dimensional quaternion covariance matrices when the quaternion samples are drawn from a population that satisfies a mild moment condition. We also apply the result to several common models.

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