Convergence in $\chi^2$ Distance to the Normal Distribution for Sums of Independent Random Variables

Vytas Zacharovas

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arXiv · 2503.21320

Convergence in $\chi^2$ Distance to the Normal Distribution for Sums of Independent Random Variables

Vytas Zacharovas

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Abstract

Suppose $n$ independent random variables $X_1, X_2, \dots, X_n$ have zero mean and equal variance. We prove that if the average of $\chi^2$ distances between these variables and the normal distribution is bounded by a sufficiently small constant, then the $\chi^2$ distance between their normalized sum and the normal distribution is $O(1/n)$.

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Vytas Zacharovas. 2025-03-27. Convergence in $\chi^2$ Distance to the Normal Distribution for Sums of Independent Random Variables. https://arxiv.org/abs/2503.21320

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