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Limiting spectral distribution for a class of random matrices. (English) Zbl 0614.62060

Let \(X=\{X_{ij}\), \(i,j=1,2,...\}\) be an infinite dimensional random matrix of i.i.d. entries and E \(X^ 2_{11}<\infty\). Let \(T_ p\) be a \(p\times p\) nonnegative definite random matrix independent of X, for \(p=1,2,... \). Suppose \(p^{-1}tr T^ k_ p\to H_ k\) a.s. as \(p\to \infty\) for \(k=1,2,...\), and \(\sum H_{2k}^{-(2k)^{-1}}<\infty\). Then the spectral distribution of \(A_ p=n^{-1}X_ pX_ p'T_ p\), where \(X_ p=\{X_{ij}\), \(i=1,...,p\), \(j=1,...,n\}\) tends to a nonrandom limit distribution as \(p\to \infty\), \(n\to \infty\), but \(p/n\to y>0.\)
The two-stage truncation method was used in proving the main result. The notion of M-graphs was introduced and some properties were derived.
Reviewer: M.Huškova

MSC:

62H10 Multivariate distribution of statistics
15B52 Random matrices (algebraic aspects)
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