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Asymptotic Products of Independent Gaussian Random Matrices with Correlated Entries

Dublin Core		PKP Metadata Items	Metadata for this Document
1.	Title	Title of document	Asymptotic Products of Independent Gaussian Random Matrices with Correlated Entries
2.	Creator	Author's name, affiliation, country	Gabriel H Tucci; Bell Labs, Alcatel-Lucent
3.	Subject	Discipline(s)
3.	Subject	Keyword(s)	Random Matrices; Limit Measures; Lyapunov Exponents; MIMO systems
3.	Subject	Subject classification	15B52; 60B20; 46L54
4.	Description	Abstract	In this work we address the problem of determining the asymptotic spectral measure of the product of independent, Gaussian random matrices with correlated entries, as the dimension and the number of multiplicative terms goes to infinity. More specifically, let $\{X_p(N)\}_{p=1}^\infty$ be a sequence of $N\times N$ independent random matrices with independent and identically distributed Gaussian entries of zero mean and variance $\frac{1}{\sqrt{N}}$. Let $\{\Sigma(N)\}_{N=1}^\infty$ be a sequence of $N\times N$ deterministic and Hermitian matrices such that the sequence converges in moments to a compactly supported probability measure $\sigma$. Define the random matrix $Y_p(N)$ as $Y_p(N)=X_p(N)\Sigma(N)$. This is a random matrix with correlated Gaussian entries and covariance matrix $E(Y_p(N)^*Y_p(N))=\Sigma(N)^2$ for every $p\geq 1$. The positive definite $N\times N$ matrix $$ B_n^{1/(2n)} (N) := \left( Y_1^* (N) Y_2^* (N) \dots Y_n^*(N) Y_n(N) \dots Y_2(N) Y_1(N) \right)^{1/(2n)} \to \nu_n $$ converges in distribution to a compactly supported measure in $[0,\infty)$ as the dimension of the matrices $N\to \infty$. We show that the sequence of measures $\nu_n$ converges in distribution to a compactly supported measure $\nu_n \to \nu$ as $n\to\infty$. The measures $\nu_n$ and $\nu$ only depend on the measure $\sigma$. Moreover, we deduce an exact closed-form expression for the measure $\nu$ as a function of the measure $\sigma$.
5.	Publisher	Organizing agency, location
6.	Contributor	Sponsor(s)
7.	Date	(YYYY-MM-DD)	2011-07-07
8.	Type	Status & genre	Peer-reviewed Article
8.	Type	Type
9.	Format	File format	PDF
10.	Identifier	Uniform Resource Identifier	http://ecp.ejpecp.org/article/view/1635
10.	Identifier	Digital Object Identifier	10.1214/ECP.v16-1635
11.	Source	Journal/conference title; vol., no. (year)	Electronic Communications in Probability; Vol 16
12.	Language	English=en
14.	Coverage	Geo-spatial location, chronological period, research sample (gender, age, etc.)
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