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Short Communications
Canonical spectral equation
V. L. Girko National Taras Shevchenko University of Kyiv, The Faculty of Cybernetics
Abstract:
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.
Keywords:
spectral functions of random matrices, Stieltjes transform, canonical spectral equation, symmetric real-valued random matrices, Lindeberg condition, eigenvalues of a random matrix. Received: 09.04.1992
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