On a formulation of the central limit theorem

TL;DR

This paper presents a non-classical formulation of the central limit theorem for certain sequences of independent random variables, emphasizing conditions like non-singularity and uniform convergence for its validity.

Contribution

It introduces a new formulation of the CLT excluding singular sequences and establishes conditions under which the theorem holds.

Findings

The CLT holds if and only if the total variance diverges.

Uniform convergence of the variance integrals is essential.

Singular sequences are excluded from the formulation.

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.