scholarly journals Invariance principles for homogeneous sums of free random variables

Bernoulli ◽  
2014 ◽  
Vol 20 (2) ◽  
pp. 586-603 ◽  
Author(s):  
Aurélien Deya ◽  
Ivan Nourdin
Keyword(s):  
Random Variables ◽  
Invariance Principles ◽  
Free Random Variables

scholarly journals Matricially free random variables

2010 ◽  
Vol 258 (12) ◽  
pp. 4075-4121 ◽  
Author(s):  
Romuald Lenczewski
Keyword(s):  
Random Variables ◽  
Free Random Variables

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.

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