A dichotomy for CLT in total variation

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

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