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

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 $O(1\mathrm{/}n)$.