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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)].

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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