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

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 $O(1/n)$.