V-density for eigenvalues of random block matrices with independent blocks whose entries have different variances and expectations

2019 ◽  
Vol 27 (3) ◽  
pp. 161-165
Author(s):  
Vyacheslav L. Girko ◽  
L. D. Shevchuk
Keyword(s):  
Random Matrices ◽  
Block Matrices ◽  
Lindeberg Condition ◽  
Random Block ◽  
Random Block Matrices

Abstract V-density under Lindeberg condition for the independent blocks of random matrices having different variances and expectations is found.

𝑉-law for random block matrices under the generalized Lindeberg condition

2019 ◽  
Vol 27 (3) ◽  
pp. 177-198
Author(s):  
Vyacheslav L. Girko
Keyword(s):  
Random Matrices ◽  
Stochastic Matrix ◽  
Block Matrices ◽  
Lindeberg Condition ◽  
Random Block ◽  
Random Block Matrices

Abstract The V-law under generalized Lindeberg condition for the independent blocks of random matrices having double stochastic matrix of covariances and different expectations of their array is proven.

scholarly journals Random block matrices generalizing the classical Jacobi and Laguerre ensembles

2010 ◽  
Vol 101 (8) ◽  
pp. 1884-1897
Author(s):  
Matthias Guhlich ◽  
Jan Nagel ◽  
Holger Dette
Keyword(s):  
Block Matrices ◽  
Random Block ◽  
Random Block Matrices

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.

From the first rigorous proof of the Circular Law in 1984 to the Circular Law for block random matrices under the generalized Lindeberg condition

2018 ◽  
Vol 26 (2) ◽  
pp. 89-116 ◽  
Author(s):  
Vyacheslav L. Girko
Keyword(s):  
Random Matrices ◽  
Stochastic Matrix ◽  
Rigorous Proof ◽  
Circular Law ◽  
Lindeberg Condition

Abstract The Circular Law under Lindeberg’s condition for the independent blocks of random matrices having zero expectations and double stochastic matrix of covariances of their array is proven.

Largest eigenvalue of large random block matrices: A combinatorial approach

2017 ◽  
Vol 06 (02) ◽  
pp. 1750008
Author(s):  
Debapratim Banerjee ◽  
Arup Bose
Keyword(s):  
Block Structure ◽  
Extreme Points ◽  
Block Size ◽  
Relative Growth ◽  
Largest Eigenvalue ◽  
Block Matrices ◽  
Variance Formula ◽  
The Matrix ◽  
Random Block ◽  
Symmetric Block

We study the largest eigenvalue of certain block matrices where the number of blocks and the block size both increase with suitable conditions on their relative growth. In one of them, we employ a symmetric block structure with large independent Wigner blocks and in the other we have the Wigner block structure with large independent symmetric blocks. The entries are assumed to be independent and identically distributed with mean [Formula: see text] variance [Formula: see text] with an appropriate growth condition on the moments. Under our conditions the limit spectral distribution of these matrices is the standard semi-circle law. It is natural to ask if the extreme eigenvalues converge to the extreme points of its support, namely [Formula: see text]. We exhibit models where this indeed happens as well as models where the spectral norm converges to [Formula: see text]. Our proofs are based on combinatorial analysis of the behavior of the trace of large powers of the matrix.

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