The central limit theorem for random determinants

V. L. Girko

Teoriâ veroâtnostej i ee primeneniâ, Tome 26 (1981) no. 3, pp. 532-542

Résumé

Let $\Xi_n$ denotes random real $(n\times n)$-matrices. Their elements $\xi_{ij}^{(n)}$ ($i,j=1\div n$) are independent, $$ \mathbf M\xi_{ij}^{(n)}=0,\qquad\mathbf D\xi_{ij}^{(n)}=1,\qquad\mathbf M(\xi_{ij}^{(n)})^4=3. $$ If there is a number $\delta>0$ such that $$ \sup_n\sup_{1\le i,j\le n} \mathbf M|\xi_{ij}^{(n)}|^{4+\delta}<\infty $$ then $$ \lim_{n\to\infty}\mathbf P\biggl\{\frac{\ln\operatorname{det}\Xi_n^2-\ln(n-1)!}{\sqrt{2\ln n}}<x\biggr\}= \frac{1}{\sqrt{2\pi}}\int_{-\infty}^x e^{-y^2/2}\,dy. $$