Approximation of Probability Measures in Variation and Products of Random Matrices

1968 ◽  
Vol 13 (1) ◽  
pp. 65-83 ◽  
Author(s):  
V. N. Tutubalin
Keyword(s):  
Random Matrices ◽  
Probability Measures ◽  
Products Of Random Matrices ◽  
Approximation Of Probability Measures

scholarly journals Products of random matrices and derivatives on p.c.f. fractals

2008 ◽  
Vol 254 (5) ◽  
pp. 1188-1216 ◽  
Author(s):  
Anders Pelander ◽  
Alexander Teplyaev
Keyword(s):  
Random Matrices ◽  
Products Of Random Matrices

scholarly journals On the Outlying Eigenvalues of a Polynomial in Large Independent Random Matrices

2019 ◽  
Author(s):  
Serban T Belinschi ◽  
Hari Bercovici ◽  
Mireille Capitaine
Keyword(s):  
Probability Theory ◽  
Random Matrices ◽  
Empirical Distribution ◽  
Free Probability ◽  
Probability Measures ◽  
Unitary Operators ◽  
Free Probability Theory ◽  
Wigner Matrix ◽  
Asymptotic Eigenvalue ◽  
Special Case

Abstract Given a selfadjoint polynomial $P(X,Y)$ in two noncommuting selfadjoint indeterminates, we investigate the asymptotic eigenvalue behavior of the random matrix $P(A_N,B_N)$, where $A_N$ and $B_N$ are independent Hermitian random matrices and the distribution of $B_N$ is invariant under conjugation by unitary operators. We assume that the empirical eigenvalue distributions of $A_N$ and $B_N$ converge almost surely to deterministic probability measures $\mu$ and $\nu$, respectively. In addition, the eigenvalues of $A_N$ and $B_N$ are assumed to converge uniformly almost surely to the support of $\mu$ and $\nu ,$ respectively, except for a fixed finite number of fixed eigenvalues (spikes) of $A_N$. It is known that almost surely the empirical distribution of the eigenvalues of $P(A_N,B_N)$ converges to a certain deterministic probability measure $\eta \ (\textrm{sometimes denoted}\ P^\square(\mu,\nu))$ and, when there are no spikes, the eigenvalues of $P(A_N,B_N)$ converge uniformly almost surely to the support of $\eta$. When spikes are present, we show that the eigenvalues of $P(A_N,B_N)$ still converge uniformly to the support of $\eta$, with the possible exception of certain isolated outliers whose location can be determined in terms of $\mu ,\nu ,P$, and the spikes of $A_N$. We establish a similar result when $B_N$ is replaced by a Wigner matrix. The relation between outliers and spikes is described using the operator-valued subordination functions of free probability theory. These results extend known facts from the special case in which $P(X,Y)=X+Y$.

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