A central limit theorem for normed spectral functions of random Jacobi matrices. (Russian) Zbl 0577.60023

Girko, V. L.; Vasil’ev, V. V.

Teor. Veroyatn. Mat. Stat. 29, 30-34 (1983).

Normed distribution functions \(\mu_ n\) of random Jacobi matrices \(H_ n=(a_ i\delta_{ij}+b_ i\delta_{ij-1}+b_ i\delta_{ij+1})\), \(i,j=1,...,n\) are considered, the row vectors of which are assumed to be i.i.d. Because it is difficult to determine in which points x the convergence M \(\mu\) \({}_ n(x)-\mu_ n(x)\to 0\) for \(n\to \infty\) does not hold, a so called smoothed normed distribution function \({\tilde \mu}{}_ n\) is defined and represented as a sum of martingale differences. By means of this a central limit theorem for \({\tilde \mu}{}_ n(x)\) is shown.

Reviewer: C.Baldauf

MSC:

60F05	Central limit and other weak theorems
15B52	Random matrices (algebraic aspects)

Keywords:

random Jacobi matrices; sum of martingale differences; central limit theorem