scholarly journals A CLT for information-theoretic statistics of Gram random matrices with a given variance profile

2008 ◽  
Vol 18 (6) ◽  
pp. 2071-2130 ◽  
Author(s):  
Walid Hachem ◽  
Philippe Loubaton ◽  
Jamal Najim
Keyword(s):  
Random Matrices ◽  
Information Theoretic ◽  
Variance Profile

scholarly journals A CLT FOR INFORMATION-THEORETIC STATISTICS OF NON-CENTERED GRAM RANDOM MATRICES

2012 ◽  
Vol 01 (02) ◽  
pp. 1150010 ◽  
Author(s):  
WALID HACHEM ◽  
MALIKA KHAROUF ◽  
JAMAL NAJIM ◽  
JACK W. SILVERSTEIN
Keyword(s):  
Random Matrices ◽  
Radio Channel ◽  
Random Variable ◽  
Multiple Antenna ◽  
Information Theoretic ◽  
Variable Formula ◽  
Main Motivation ◽  
Unit Variance ◽  
The Moment ◽  
Variance Profile

In this article, we study the fluctuations of the random variable: [Formula: see text] where [Formula: see text], as the dimensions of the matrices go to infinity at the same pace. Matrices Xn and An are respectively random and deterministic N × n matrices; matrices Dn and [Formula: see text] are deterministic and diagonal, with respective dimensions N × N and n × n; matrix Xn = (Xij) has centered, independent and identically distributed entries with unit variance, either real or complex. We prove that when centered and properly rescaled, the random variable [Formula: see text] satisfies a Central Limit Theorem and has a Gaussian limit. The variance of [Formula: see text] depends on the moment [Formula: see text] of the variables Xij and also on its fourth cumulant [Formula: see text]. The main motivation comes from the field of wireless communications, where [Formula: see text] represents the mutual information of a multiple antenna radio channel. This article closely follows the companion article "A CLT for Information-theoretic statistics of Gram random matrices with a given variance profile", Ann. Appl. Probab. (2008) by Hachem et al., however the study of the fluctuations associated to non-centered large random matrices raises specific issues, which are addressed here.

scholarly journals Linear eigenvalue statistics of random matrices with a variance profile

2020 ◽  
pp. 2250004
Author(s):  
Kartick Adhikari ◽  
Indrajit Jana ◽  
Koushik Saha
Keyword(s):  
Random Matrices ◽  
Random Matrix ◽  
Random Variable ◽  
Second Order ◽  
Linear Eigenvalue Statistics ◽  
Eigenvalue Statistics ◽  
Eigenvalue Statistic ◽  
Variation Distance ◽  
Random Matrix Ensembles ◽  
Variance Profile

We give an upper bound on the total variation distance between the linear eigenvalue statistic, properly scaled and centered, of a random matrix with a variance profile and the standard Gaussian random variable. The second-order Poincaré inequality-type result introduced in [S. Chatterjee, Fluctuations of eigenvalues and second order poincaré inequalities, Prob. Theory Rel. Fields 143(1) (2009) 1–40.] is used to establish the bound. Using this bound, we prove central limit theorem for linear eigenvalue statistics of random matrices with different kind of variance profiles. We re-establish some existing results on fluctuations of linear eigenvalue statistics of some well-known random matrix ensembles by choosing appropriate variance profiles.

Some patterned matrices with independent entries

2020 ◽  
pp. 2150030
Author(s):  
Arup Bose ◽  
Koushik Saha ◽  
Priyanka Sen
Keyword(s):  
Random Matrices ◽  
Spectral Distribution ◽  
Finite Variance ◽  
Weighted Sum ◽  
Hankel Matrices ◽  
Special Cases ◽  
Common Thread ◽  
Different Types ◽  
Patterned Matrices ◽  
Variance Profile

Patterned random matrices such as the reverse circulant, the symmetric circulant, the Toeplitz and the Hankel matrices and their almost sure limiting spectral distribution (LSD), have attracted much attention. Under the assumption that the entries are taken from an i.i.d. sequence with finite variance, the LSD are tied together by a common thread — the [Formula: see text]th moment of the limit equals a weighted sum over different types of pair-partitions of the set [Formula: see text] and are universal. Some results are also known for the sparse case. In this paper, we generalize these results by relaxing significantly the i.i.d. assumption. For our models, the limits are defined via a larger class of partitions and are also not universal. Several existing and new results for patterned matrices, their band and sparse versions, as well as for matrices with continuous and discrete variance profile follow as special cases.

scholarly journals Non-Hermitian random matrices with a variance profile (I): deterministic equivalents and limiting ESDs

2018 ◽  
Vol 23 (0) ◽  
Author(s):  
Nicholas Cook ◽  
Walid Hachem ◽  
Jamal Najim ◽  
David Renfrew
Keyword(s):  
Random Matrices ◽  
Variance Profile

Large-sample confidence intervals of information-theoretic measures in linguistics

2020 ◽  
Vol 6 (1) ◽  
pp. 19-54
Author(s):  
Ryan Ka Yau Lai ◽  
Youngah Do
Keyword(s):  
Maximum Likelihood ◽  
Corpus Linguistics ◽  
Delta Method ◽  
Confidence Bounds ◽  
Likelihood Estimator ◽  
Information Theoretic ◽  
Leibler Divergence ◽  
Information Theoretic Measures ◽  
Data Points ◽  
Measure Of Uncertainty

This article explores a method of creating confidence bounds for information-theoretic measures in linguistics, such as entropy, Kullback-Leibler Divergence (KLD), and mutual information. We show that a useful measure of uncertainty can be derived from simple statistical principles, namely the asymptotic distribution of the maximum likelihood estimator (MLE) and the delta method. Three case studies from phonology and corpus linguistics are used to demonstrate how to apply it and examine its robustness against common violations of its assumptions in linguistics, such as insufficient sample size and non-independence of data points.

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