An equation for the Stieltjes transform of the limit spectral functions of isotropic random matrices. (Russian) Zbl 0644.60055

Girko, V. L.

Teor. Veroyatn. Mat. Stat., Kiev 36, 31-33 (1987).

Let \(S=({\vec \xi}_ 1,...,{\vec \xi}_ n)\), \({\vec \xi}_ i\approx {\vec \xi}\), be an isotropic random matrix, \(\lambda_ k\) be eigenvalues of the matrix \(n^{-1}SS'\), \(\mu_ n(x)=n^{-1}\sum^{n}_{1}F(x- \lambda_ k)\), \(F(y)=0\) if \(y\leq 0\) and \(F(y)=1\) if \(y>0.\)
Theorem. If for some m and n the vectors \(\xi_ i\) are independent and \(\lim_{n}m n^{-1}=c\), \(0<c<\infty\), \[ \lim_{n}P\{({\vec \xi},{\vec \xi})m^{-1}<x\}=G(x),\quad E \xi^ 4_ 1<a<\infty \text{ then } P- \lim_{n}\mu_ n(x)=\mu (x), \] and if \(u(t)=\int^{\infty}_{0}(t+x)^{-1}d\mu (x)\), then \[ u(t)=\int^{\infty}_{0}[t+x(1+x(u(t)))^{-1}c]^{-1}dG(x),\quad t>0. \]

Reviewer: G.A.Sokhadze

MSC:

60H25	Random operators and equations (aspects of stochastic analysis)
60F99	Limit theorems in probability theory
15B52	Random matrices (algebraic aspects)

Keywords:

isotropic random matrix