## 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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### References:

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