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

We estimate the L\'evy 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≥5. The proof relies on an inequality that controls the L\'evy distance to the Bernoulli law by the values of the reciprocal Cauchy transform at two points. For each r≥2, 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.