On the eigenvalues of random matrices

1994 ◽  
Vol 31 (A) ◽  
pp. 49-62 ◽  
Author(s):  
Persi Diaconis ◽  
Mehrdad Shahshahani
Keyword(s):  
Random Matrices ◽  
Haar Measure ◽  
Random Matrix ◽  
Unitary Group ◽  
Symmetric Groups ◽  
Random Variable ◽  
Normal Random Variable ◽  
Eigenvalues Of Random Matrices

Let M be a random matrix chosen from Haar measure on the unitary group Un. Let Z = X + iY be a standard complex normal random variable with X and Y independent, mean 0 and variance ½ normal variables. We show that for j = 1, 2, …, Tr(Mj) are independent and distributed as √jZ asymptotically as n →∞. This result is used to study the set of eigenvalues of M. Similar results are given for the orthogonal and symplectic and symmetric groups.

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.

Generation of Correlated Logistic-Normal Random Variates for Medical Decision Trees

1998 ◽  
Vol 37 (03) ◽  
pp. 235-238 ◽  
Author(s):  
M. El-Taha ◽  
D. E. Clark
Keyword(s):  
Monte Carlo ◽  
Decision Trees ◽  
Computer Algebra ◽  
Taylor Series ◽  
Computer Algebra System ◽  
Random Variable ◽  
Monte Carlo Analysis ◽  
Normal Random Variable ◽  
Medical Decision ◽  
Theoretical Results

AbstractA Logistic-Normal random variable (Y) is obtained from a Normal random variable (X) by the relation Y = (ex)/(1 + ex). In Monte-Carlo analysis of decision trees, Logistic-Normal random variates may be used to model the branching probabilities. In some cases, the probabilities to be modeled may not be independent, and a method for generating correlated Logistic-Normal random variates would be useful. A technique for generating correlated Normal random variates has been previously described. Using Taylor Series approximations and the algebraic definitions of variance and covariance, we describe methods for estimating the means, variances, and covariances of Normal random variates which, after translation using the above formula, will result in Logistic-Normal random variates having approximately the desired means, variances, and covariances. Multiple simulations of the method using the Mathematica computer algebra system show satisfactory agreement with the theoretical results.

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.

Improved approximate confidence intervals for the mean of a log-normal random variable

2002 ◽  
Vol 21 (10) ◽  
pp. 1443-1459 ◽  
Author(s):  
Douglas J. Taylor ◽  
Lawrence L. Kupper ◽  
Keith E. Muller
Keyword(s):  
Confidence Intervals ◽  
Random Variable ◽  
Normal Random Variable ◽  
The Mean ◽  
Approximate Confidence Intervals ◽  
Log Normal

scholarly journals Occupation times of alternating renewal processes with Lévy applications

2018 ◽  
Vol 55 (4) ◽  
pp. 1287-1308 ◽  
Author(s):  
Nicos Starreveld ◽  
Réne Bekker ◽  
Michel Mandjes
Keyword(s):  
Central Limit Theorem ◽  
Limit Theorem ◽  
Central Limit ◽  
Random Variable ◽  
Levy Processes ◽  
Normal Random Variable ◽  
Occupation Time ◽  
Renewal Processes ◽  
Tail Asymptotics ◽  
Occupation Times

AbstractIn this paper we present a set of results relating to the occupation time α(t) of a processX(·). The first set of results concerns exact characterizations of α(t), e.g. in terms of its transform up to an exponentially distributed epoch. In addition, we establish a central limit theorem (entailing that a centered and normalized version of α(t)∕tconverges to a zero-mean normal random variable ast→∞) and the tail asymptotics of ℙ(α(t)∕t≥q). We apply our findings to spectrally positive Lévy processes reflected at the infimum and establish various new occupation time results for the corresponding model.

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