A dichotomy for CLT in total variation

TL;DR

This paper establishes a clear dichotomy in the behavior of the total variation distance between sums of i.i.d. variables and the normal distribution, showing it is either always maximal or decays at a specific rate.

Contribution

It proves a dichotomy result for the total variation distance in the CLT, identifying two distinct regimes of convergence behavior.

Findings

If the total variation distance is not always 1, it decreases at a rate of n^{-1/2}.

The total variation distance is either constantly 1 or converges to zero at a specific rate.

The result applies to i.i.d. variables with finite third moments.

Abstract

Let $\eta_i$, $i\ge 1$, be a sequence of independent and identically distributed random variables with finite third moment, and let $\Delta_n$ be the total variation distance between the distribution of $S_n:=\sum_{i=1}^n\eta_i$ and the normal distribution with the same mean and variance. In this note, we show the dichotomy that either $\Delta_n=1$ for all $n$ or $\Delta_n=O\left(n^{-1/2}\right)$.