scholarly journals Gaussian Fluctuations for Linear Eigenvalue Statistics of Products of Independent iid Random Matrices

2019 ◽  
Vol 33 (3) ◽  
pp. 1541-1612
Author(s):  
Natalie Coston ◽  
Sean O’Rourke
Keyword(s):  
Random Matrices ◽  
Linear Eigenvalue Statistics ◽  
Gaussian Fluctuations ◽  
Eigenvalue Statistics

On the fluctuations of eigenvalues of multiplicative deformed unitary invariant ensembles

2016 ◽  
Vol 05 (02) ◽  
pp. 1650007 ◽  
Author(s):  
Vladimir Vasilchuk
Keyword(s):  
Central Limit Theorem ◽  
Limit Theorem ◽  
Random Matrices ◽  
Central Limit ◽  
Counting Measure ◽  
Linear Eigenvalue Statistics ◽  
The Central Limit Theorem ◽  
Eigenvalue Statistics ◽  
Leading Term

We consider the ensemble of [Formula: see text] random matrices [Formula: see text], where [Formula: see text] and [Formula: see text] are non-random, unitary, having the limiting Normalized Counting Measure (NCM) of eigenvalues, and [Formula: see text] is unitary, uniformly distributed over [Formula: see text]. We find the leading term of the covariance of traces of resolvent of [Formula: see text] and establish the Central Limit Theorem for sufficiently smooth linear eigenvalue statistics of [Formula: see text] as [Formula: see text].

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.

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