TY - UNPB

T1 - On a formulation of the central limit theorem

AU - Шмыров, Александр Сергеевич

AU - Шмыров, Василий Александрович

N1 - 6 pages

PY - 2025/1/28

Y1 - 2025/1/28

N2 - 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.

AB - 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.

KW - math.PR

KW - 60F05

KW - G.3

M3 - Preprint

BT - On a formulation of the central limit theorem

ER -