Some remarks on summing independent variables in the non-classical case

V. I. Rotar'

Teoriâ veroâtnostej i ee primeneniâ, Tome 21 (1976) no. 1, pp. 128-135

Résumé

Let $\{F_{jn}\}_{j=1}^n$ and $\{G_{jn}\}_{j=1}^n$ be two triangular arrays of distribution functions. Let $b_n$ be such that $$ \sum_{j=1}^nb_n^{-2}\int_0^{b_n}x[1-F_{jn}(x)+F_{jn}(-x)]\,dx=\delta/n, $$ where $0\delta\le 1$; $$ a_{jn}=\int_{-b_n}^{b_n}x\,dF_{jn}(x),\qquad a'_{jn}=\int_{-b_n}^{b_n}x\,dG_{jn}(x). $$ The paper deals with conditions under which $$ *\hskip-4,5mm\prod_{j=1}^nF_{jn}(xb_n+a_{jn})-{}{*\hskip-4,5mm}\prod_{j=1}^nG_{jn}(xb_n+a'_{jn})\to 0 $$ weakly with respect to the classes $C$ or $C_0$.