A dichotomy for CLT in total variation

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 $\Delta_n=O\left(n^{-1/2}\right)$.