scholarly journals FLUCTUATIONS OF MATRIX ENTRIES OF ANALYTIC FUNCTIONS OF NON-HERMITIAN RANDOM MATRICES

2012 ◽  
Vol 01 (03) ◽  
pp. 1250008
Author(s):  
SEAN O'ROURKE
Keyword(s):  
Random Matrices ◽  
Random Matrix ◽  
Analytic Functions ◽  
Random Variables

Consider an n × n non-Hermitian random matrix Mn whose entries are independent real random variables. Under suitable conditions on the entries, we study the fluctuations of the entries of f(Mn) as n tends to infinity, where f is analytic on an appropriate domain. This extends the results in [19, 20, 23] from symmetric random matrices to the non-Hermitian case.

Sparse signal recovery using a new class of random matrices

2017 ◽  
Vol 8 (2) ◽  
Author(s):  
Enrico Au-Yeung
Keyword(s):  
Random Matrices ◽  
Random Matrix ◽  
Sparse Matrices ◽  
Random Variables ◽  
Signal Recovery ◽  
Original Matrix ◽  
Bernoulli Random Variables ◽  
New Class ◽  
The Matrix ◽  
New Type

Abstract We propose a new class of random matrices that enables the recovery of signals with sparse representation in a known basis with overwhelmingly high probability. To construct a matrix in this class, we begin with a fixed non-random matrix that satisfies two very general conditions. Then we decompose the matrix into pieces of sparse matrices. A random sum (involving Bernoulli random variables) of these pieces of sparse matrices is used to construct the final matrix. We say that the random matrix is the randomized Bernoulli transform of the original matrix. The random matrix is not created by filling all its entries with random variables, as in the case of Gaussian or Bernoulli matrices. Therefore, as a benefit, far fewer number of random variables are needed to generate this new type of random matrices. We prove that the number of samples needed to recover a random signal is proportional to the sparsity of the signal, up to a logarithmic factor, and hence this number is nearly optimal.

VICTORIA transform, RESPECT and REFORM methods for the proof of the G-Elliptic Law under G-Lindeberg condition and twice stochastic condition for the variances and covariances of the entries of some random matrices

2020 ◽  
Vol 28 (2) ◽  
pp. 131-162
Author(s):  
Vyacheslav L. Girko
Keyword(s):  
Random Matrices ◽  
Random Matrix ◽  
Lindeberg Condition

AbstractThe G-Elliptic law under the G-Lindeberg condition for the independent pairs of the entries of a random matrix is proven.

Fourier Analysis and Benfordʼs Law

Benford's Law ◽  
2015 ◽  
Author(s):  
Steven J. Miller
Keyword(s):  
Distribution Function ◽  
Fourier Analysis ◽  
Random Matrix ◽  
Matrix Theory ◽  
Random Variables ◽  
Cumulative Distribution ◽  
Benford’S Law ◽  
Good System ◽  
Benford's Law ◽  
Geometric Brownian Motions

This chapter continues the development of the theory of Benford's law. It uses Fourier analysis (in particular, Poisson Summation) to prove many systems either satisfy or almost satisfy the Fundamental Equivalence, and hence either obey Benford's law, or are well approximated by it. Examples range from geometric Brownian motions to random matrix theory to products and chains of random variables to special distributions. The chapter furthermore develops the notion of a Benford-good system. Unfortunately one of the conditions here concerns the cancelation in sums of translated errors related to the cumulative distribution function, and proving the required cancelation often requires techniques specific to the system of interest.

scholarly journals Heavy-tailed random matrices

2018 ◽  
Author(s):  
Vladimir Kravtsov
Keyword(s):  
Random Matrices ◽  
Heavy Tails ◽  
Probability Distributions ◽  
Basin Of Attraction ◽  
Random Variables ◽  
Universal Properties ◽  
Random Matrix Ensembles ◽  
Free Random Variables ◽  
Heavy Tailed ◽  
Non Gaussian

This article considers non-Gaussian random matrices consisting of random variables with heavy-tailed probability distributions. In probability theory heavy tails of distributions describe rare but violent events which usually have a dominant influence on the statistics. Furthermore, they completely change the universal properties of eigenvalues and eigenvectors of random matrices. This article focuses on the universal macroscopic properties of Wigner matrices belonging to the Lévy basin of attraction, matrices representing stable free random variables, and a class of heavy-tailed matrices obtained by parametric deformations of standard ensembles. It first examines the properties of heavy-tailed symmetric matrices known as Wigner–Lévy matrices before discussing free random variables and free Lévy matrices as well as heavy-tailed deformations. In particular, it describes random matrix ensembles obtained from standard ensembles by a reweighting of the probability measure. It also analyses several matrix models belonging to heavy-tailed random matrices and presents methods for integrating them.

scholarly journals On the limiting spectral density of random matrices filled with stochastic processes

2019 ◽  
Vol 27 (2) ◽  
pp. 89-105 ◽  
Author(s):  
Matthias Löwe ◽  
Kristina Schubert
Keyword(s):  
Stochastic Process ◽  
Spectral Density ◽  
Random Matrices ◽  
Markov Processes ◽  
Random Matrix ◽  
Matrix Theory ◽  
Gibbs Measures ◽  
State Spaces ◽  
The Matrix ◽  
Finite State

Abstract We discuss the limiting spectral density of real symmetric random matrices. In contrast to standard random matrix theory, the upper diagonal entries are not assumed to be independent, but we will fill them with the entries of a stochastic process. Under assumptions on this process which are satisfied, e.g., by stationary Markov chains on finite sets, by stationary Gibbs measures on finite state spaces, or by Gaussian Markov processes, we show that the limiting spectral distribution depends on the way the matrix is filled with the stochastic process. If the filling is in a certain way compatible with the symmetry condition on the matrix, the limiting law of the empirical eigenvalue distribution is the well-known semi-circle law. For other fillings we show that the semi-circle law cannot be the limiting spectral density.

scholarly journals The Expected Norm of Random Matrices

2000 ◽  
Vol 9 (2) ◽  
pp. 149-166 ◽  
Author(s):  
YOAV SEGINER
Keyword(s):  
Random Matrices ◽  
Random Matrix ◽  
Operator Norm ◽  
Euclidean Norm ◽  
Constant Independent ◽  
Multiplicative Constant ◽  
The Matrix

We compare the Euclidean operator norm of a random matrix with the Euclidean norm of its rows and columns. In the first part of this paper, we show that if A is a random matrix with i.i.d. zero mean entries, then E∥A∥h [les ] Kh (E maxi ∥ai[bull ] ∥h + E maxj ∥aj[bull ] ∥h), where K is a constant which does not depend on the dimensions or distribution of A (h, however, does depend on the dimensions). In the second part we drop the assumption that the entries of A are i.i.d. We therefore consider the Euclidean operator norm of a random matrix, A, obtained from a (non-random) matrix by randomizing the signs of the matrix's entries. We show that in this case, the best inequality possible (up to a multiplicative constant) is E∥A∥h [les ] (c log1/4 min {m, n})h (E maxi ∥ai[bull ] ∥h + E maxj ∥aj[bull ] ∥h) (m, n the dimensions of the matrix and c a constant independent of m, n).

A Block Compressive Sensing Based Scalable Encryption Framework for Protecting Significant Image Regions

2016 ◽  
Vol 26 (11) ◽  
pp. 1650191 ◽  
Author(s):  
Yushu Zhang ◽  
Jiantao Zhou ◽  
Fei Chen ◽  
Leo Yu Zhang ◽  
Di Xiao ◽  
...  
Keyword(s):  
Compressive Sensing ◽  
Random Matrices ◽  
Random Matrix ◽  
Chaotic Maps ◽  
Sampling Rate ◽  
Equal Size ◽  
Edge Detector ◽  
Scalable Encryption ◽  
Significant Block ◽  
Block Encryption

The existing Block Compressive Sensing (BCS) based image ciphers adopted the same sampling rate for all the blocks, which may lead to the desirable result that after subsampling, significant blocks lose some more-useful information while insignificant blocks still retain some less-useful information. Motivated by this observation, we propose a scalable encryption framework (SEF) based on BCS together with a Sobel Edge Detector and Cascade Chaotic Maps. Our work is firstly dedicated to the design of two new fusion techniques, chaos-based structurally random matrices and chaos-based random convolution and subsampling. The basic idea is to divide an image into some blocks with an equal size and then diagnose their respective significance with the help of the Sobel Edge Detector. For significant block encryption, chaos-based structurally random matrix is applied to significant blocks whereas chaos-based random convolution and subsampling are responsible for the remaining insignificant ones. In comparison with the BCS based image ciphers, the SEF takes lightweight subsampling and severe sensitivity encryption for the significant blocks and severe subsampling and lightweight robustness encryption for the insignificant ones in parallel, thus better protecting significant image regions.

scholarly journals Identifiability of parametric random matrix models

2019 ◽  
Vol 22 (03) ◽  
pp. 1950018
Author(s):  
Tomohiro Hayase
Keyword(s):  
Probability Theory ◽  
Matrix Models ◽  
Random Matrices ◽  
Random Matrix ◽  
Free Probability ◽  
Parameter Identifiability ◽  
Free Probability Theory ◽  
Noise Type ◽  
Random Matrix Models ◽  
Spectral Distributions

We investigate parameter identifiability of spectral distributions of random matrices. In particular, we treat compound Wishart type and signal-plus-noise type. We show that each model is identifiable up to some kind of rotation of parameter space. Our method is based on free probability theory.

scholarly journals Permutations Without Long Decreasing Subsequences and Random Matrices

10.37236/929 ◽  
2007 ◽  
Vol 14 (1) ◽  
Author(s):  
Piotr Šniady
Keyword(s):  
Random Matrices ◽  
Young Diagram ◽  
Random Matrix ◽  
Random Permutation ◽  
Fixed Number ◽  
Plancherel Measure ◽  
Young Diagrams ◽  
Gaussian Unitary Ensemble ◽  
Longest Increasing Subsequence ◽  
Unitary Ensemble

We study the shape of the Young diagram $\lambda$ associated via the Robinson–Schensted–Knuth algorithm to a random permutation in $S_n$ such that the length of the longest decreasing subsequence is not bigger than a fixed number $d$; in other words we study the restriction of the Plancherel measure to Young diagrams with at most $d$ rows. We prove that in the limit $n\to\infty$ the rows of $\lambda$ behave like the eigenvalues of a certain random matrix (namely the traceless Gaussian Unitary Ensemble random matrix) with $d$ rows and columns. In particular, the length of the longest increasing subsequence of such a random permutation behaves asymptotically like the largest eigenvalue of the corresponding random matrix.

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