The quarter-of-circumference law. (English. Russian original) Zbl 0873.60024

Girko, V. L.; Repin, K. Yu.

Theory Probab. Math. Stat. 50, 67-70 (1995); translation from Teor. Jmovirn. Mat. Stat. 50, 66-69 (1994).

The authors consider an asymmetric matrix \((\xi_{ij})_{i,j=1}^n\), where the elements \(\xi_{ij}\) are independent, \(E \xi_{ij}=0\), \(\text{Var }\xi_{ij}=n^{-1}\sigma^2\), \(0\leq \sigma\leq C<\infty\) and the Lindeberg condition is satisfied, and proved that for \(V_n(x)=n^{-1}\sum_1^n\chi(\sqrt {\lambda_k}<x)\), \[ p\lim_{n\to\infty}V_n(x)= \begin{cases} 0, &x<0,\\ \pi^{-1}\sigma^{-2} \int_0^x (4\sigma^2-u^2)^{1/2} du, &0\leq x\leq 2\pi,\\ 1, &x>2\pi, \end{cases} \] i.e., the limit distribution density is a quarter of a circumference.

Reviewer: N.Renganathan (Annamalai Nagar)

MSC:

60F99	Limit theorems in probability theory
60E99	Distribution theory

Keywords:

random matrix; eigenvalues; spectral functions