On a formulation of the central limit theorem

Alexander Shmyrov; Vasily Shmyrov

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arXiv · 2501.16963

On a formulation of the central limit theorem

Alexander Shmyrov·Vasily Shmyrov

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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.

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Alexander Shmyrov, Vasily Shmyrov. 2025-01-28. On a formulation of the central limit theorem. https://arxiv.org/abs/2501.16963

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