Canonical spectral equation

V. L. Girko

Teoriâ veroâtnostej i ee primeneniâ, Tome 39 (1994) no. 4, pp. 812-820

Résumé

We consider a sequence of symmetric real-valued random matrices $\Xi _n = (\xi _{ij}^{(n)} )_{i,j = 1}^n ,n = 1,2, \ldots $, whose entries $\xi _{ij}^{(n)}$ ,$i \ge j$,$i,j = 1, \ldots ,n,$, are independent for each $n$, whereas $\mathbf{E}\xi _{ij}^{(n)} = a_{ij}^{(n)} ,\operatorname{Var}\xi _{ij}^{(n)} = \sigma _{ij}^{(n)}$, $i \ge j$, $i,j = 1, \ldots ,n,$ $$ \sup_n\max_{i = 1, \ldots ,n} \sum_{j = 1}^n {\sigma _{ij}^{(n)} \infty} ,\qquad \sup_n\max_{i = 1, \ldots ,n} \sum_{j = 1}^n {| {a_{ij}^{(n)} }| \infty ,} $$ and the Lindeberg condition is satisfied for these entries: for any $\tau > 0$, $$ \lim_{n \to \infty }\max _{i = 1, \ldots ,n} \sum_{j = 1}^n {\mathbf{E}[ {\xi _{ij}^{(n)} - a_{ij}^{(n)} } ]^2 \chi \{ {|\xi _{ij}^{(n)} - a_{ij}^{(n)} | > \tau }\} = 0.} $$ We prove that $p\lim _{n \to \infty } \sup _x |\mu _n (x) - F_n (x)| = 0$, where $\mu _n (x) = n^{ - 1} \Sigma _{k = 1}^n \chi (\omega :\lambda _k x),\lambda _1 \ge \cdots \ge \lambda _n $ are the eigenvalues of the random matrix $\Xi _n = (\xi _{ij}^{(n)} )_{i,j = 1}^n ,F_n (x)$ are distribution functions, the Stieltjes transforms of which are equal to $$ \int {(x - z)^{ - 1} dF_n (x) = n^{ - 1} \sum_{i = 1}^n {c_i (z),\quad z = t + is,\quad s \ne 0,} } $$ and the functions $c_i (z)$ satisfy the system of equations $$ c_i (z) = \left\{ {\left[ {A - zI_n - \delta _{pl} \sum_{s = 1}^n {c_s (z)\sigma _{sl}^{(n)} } } \right]^{ - 1} } \right\}_{ii} ,\quad i = 1, \ldots ,n, $$ where $\delta _{pl} $ is the Kronecker symbol, $A_n = (a_{ij}^{(n)} )_{i,j = 1}^n ,I_n $ is the identity matrix of the $n$th order.

Détail

Keywords: spectral functions of random matrices, canonical spectral equation, symmetric real-valued random matrices, Lindeberg condition, eigenvalues of a random matrix.
Mots-clés : Stieltjes transform