An extension of the Lindeberg-Feller theorem

A new version is presented of the necessary and sufficient condition of convergence, to a normal law, of sums of independent variables in a nonclassical situation (i.e., absence of limiting negligibility of variables). The obtained condition differs from previously obtained conditions by the fact that it does not use Levy's metric and that is is closer to classical formulations. A similar condition is sufficient for the closeness of two convolutions when the number of components of the convolutions increases without bounds.