On a formulation of the central limit theorem

Abstract: A non-classical formulation of the central limit theorem is given for sequences of independent random variables with finite second moments. Singular sequences whose members all have a degenerate or normal distribution are excluded from consideration. The condition of uniform convergence is imposed on the improper integrals defining the variances. Under the conditions of non-singularity and uniform convergence, the central limit theorem is valid if and only if the total variance increases indefinitely.