A limit theorem of the type of the law of large numbers for random determinants. (English. Russian original) Zbl 0749.60019

Akulov, G. V.

Theory Probab. Math. Stat. 43, 1-4 (1991); translation from Teor. Veroyatn. Mat. Stat., Kiev 43, 3-6 (1990).

For every \(n\geq 1\) let \(A_ n=(x^ n_{ij})\) be a random \(n\times n\)- matrix, whose elements \(x^ n_{ij}\) are independent and have a density \(p^ n_{ij}\) relative to the Lebesgue measure. Define \(y_ n=\ln|\text{det} A_ n|\). Then the weak law of large numbers is proved for the sequence \((y_ n)\) under the following two conditions: (1) \(\sup_{n,i,j}\int p^ n_{ij}(x)^{1+c}dx<\infty\) for some \(c>0\), and (2) \(\sup_{n,i,j}\mathbb{E}(x^ n_{i,j})^ 2<\infty\).

Reviewer: E.Dettweiler (Reutlingen)

MSC:

60F05

Central limit and other weak theorems

Keywords:

random matrix; determinant; weak law of large numbers