## Strong convergence of the empirical distribution of eigenvalues of large dimensional random matrices.(English)Zbl 0851.62015

Let $$X_n = [X^n_{ij}]$$, $$1 \leq i \leq n$$, $$1 \leq j \leq N$$, and $$T_n = [T^n_{ij}]$$, $$1 \leq i,j \leq n$$, be random matrices given on a common probability space, where $$X_n$$ contains identically distributed entries $$X^n_{ij} \in \mathbb{C}$$ for all $$n$$, $$i$$, $$j$$, independent across $$i$$, $$j$$ for each $$n \geq 1$$ with $$E |X_{11}^1 - EX^1_{11} |^2 = 1$$, and $$T_n$$ is Hermitian nonnegative definite, with empirical distribution function (e.d.f.) $$F^{T_n}$$ of (only real) eigenvalues of $$T_n$$ converging a.s. in distribution to a probability distribution function (p.d.f.) $$H$$ on $$[0,\infty)$$ as $$n \to \infty$$. Furthermore, let $$X_n$$ and $$T_n$$ be independent, and $$N = N(n)$$ satisfy $$n/N \to c > 0$$ as $$n \to \infty$$.
The main theorem proved in the paper states that under the above assumptions the e.d.f. $$F^{(1/N)X_n X^*_n T_n}$$ of the eigenvalues of $$(1/N)X_n X^*_n T_n$$ converges a.s. in distribution to a nonrandom p.d.f. $$F$$, whose Stieltjes transform $$m = m(z)$$, $$z \in \mathbb{C}^+ = \{z \in \mathbb{C} : \text{Im } z > 0\}$$, is the unique solution to $m = \int [\tau(1 - c - czm)-z]^{-1} dH (\tau)$ in the set $$\{m \in \mathbb{C} : - (1-c)/z + cm \in \mathbb{C}^+\}$$. The obtained result extends to the complex case the limit theorem of Y. Q. Yin [ibid. 20, 50-68 (1986; Zbl 0614.62060)].

### MSC:

 62E20 Asymptotic distribution theory in statistics 62G30 Order statistics; empirical distribution functions 60F15 Strong limit theorems 62H99 Multivariate analysis 15B52 Random matrices (algebraic aspects)

Zbl 0614.62060
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