A generalization of the Lindeberg–Feller theorem

Let $\xi_n=\xi_{n1}+\dots+\xi_{nj}\dots$ be a sequence of sums of independent random variables with a finite or infinite number of summands. Suppose that
$$ \mathbf E\xi_{nj}=0\quad\sigma_{nj}^2=\mathbf E\xi_{nj}^2<\infty\quad\sum_j\sigma_{nj}^2=1 $$
and denote
\begin{gather*} F_n(x)=\mathbf P\{\xi_n<x\},\quad F_{nj}(x)=\mathbf P\{\xi_{nj}<x\}, \\ \Phi(x)=\frac1{\sqrt{2\pi}}\int_{-\infty}^x\exp(-t^2/2)\,dt,\quad \Phi_{nj}(x)=\Phi(x/\sigma_{nj}). \end{gather*}

In the present paper the following theorem is proved: {\it for $\sup\limits_x|F_n(x)-\Phi(x)|\to0$ as $n\to\infty$ thе necessary and sufficient conditions are
$1^\circ\ \sup\limits_jL(F_{nj},\Phi_{nj})\to0$ ($L$ is the Lévy metric);
$2^\circ$ for every positive $\varepsilon$
$$ \sum_j\int_{|x|\ge\varepsilon}x^2d(F_{nj}-\Phi_{nj})\to0. $$
}