A refinement of the central limit theorem for random determinants

V. L. Girko

Teoriâ veroâtnostej i ee primeneniâ, Tome 42 (1997) no. 1, pp. 63-73

Résumé

The paper proves the central limit theorem (the logarithmic law) for random determinants under weaker conditions than the author used earlier: if for any $n$ the random elements $\xi^{(n)}_{ij}$, $i,j=1,\dots,n$, of the matrix $\Xi=(\xi_{ij}/n)$ are independent, $\mathsf{E}\xi_{ij}^{(n)}=a$, $\operatorname{Var}\xi_{ij}^{(n)}=1$, and for some $\delta > 0$ $$ \sup_n\max_{i,j=1,\dots,n}\mathsf{E}|\xi_{ij}^{(n)}|^{4+\delta}<\infty, $$ then \begin{align*} &\lim_{n\to\infty}\biggl\{\frac{\log\det\Xi^2-\log(n-1)!\,-\log(1+na^2)}{\sqrt{2\log n}}<x\biggr\} \\ &\qquad=\frac1{\sqrt{2\pi}}\int_{-\infty}^x\exp\biggl(-\frac{u^2}2\biggr)\,du. \end{align*}

Détail

Keywords: logarithmic law, random determinants, method of perpendiculars, normal regularization (regularity).