The Berry--Esseen estimate in the Boolean central limit theorem

Abstract

We estimate the Lévy distance to the symmetric Bernoulli distribution for normalized Boolean convolution powers of a probability measure with mean zero and variance one. If the absolute moment of order $r$ is finite, we obtain the upper bounds $o(n^{-(r-2)/6})$ for $2<r<5$ and $O(n^{-1/2})$ for $r\ge5$. The proof relies on an inequality that controls the Lévy distance to the Bernoulli law by the values of the reciprocal Cauchy transform at two points. For each $r\ge2$, we also construct a single fixed distribution whose absolute moments are finite exactly up to order $r$, and show that the exponent $\min\{(r-2)/6,1/2\}$ cannot be improved. Under only a finite second moment assumption, the convergence can be arbitrarily slow.