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

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 $C_0$.