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Strong Convergence of the Empirical Distribution of Eigenvalues of Large Dimensional Random Matrices

โœ Scribed by J.W. Silverstein

Publisher
Elsevier Science
Year
1995
Tongue
English
Weight
267 KB
Volume
55
Category
Article
ISSN
0047-259X

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โœฆ Synopsis

Let (X) be (n \times N) containing i.i.d. complex entries with (\mathbf{E}\left|X_{11}-\mathbf{E} X_{11}\right|^{2}=1), and (T) an (n \times n) random Hermitian nonnegative definite, independent of (X). Assume, almost surely, as (n \rightarrow \infty), the empirical distribution function (e.d.f.) of the eigenvalues of (T) converges in distribution, and the ratio (n / N) tends to a positive number. Then it is shown that, almost surely, the e.d.f. of the eigenvalues of ((1 / N) X X^{*} T) converges in distribution. The limit is nonrandom and is characterized in terms of its Stieltjes transform, which satisfies a certain equation. (c) 1995 Academic Press. Inc.

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