A note on the rate of convergence in the Boolean central limit theorem

Several results giving upper bounds for the speed of convergence in non-commutative central limit theorems are known. A natural question is whether there also exist lower bounds, or whether there exist classes of probability measures where the speed of convergence is faster. In this paper, we answer this for the Boolean central limit theorem in the bounded support case and using the Levy distance. We show that for a probability measure mu of bounded support and nonzero third moment the rate of convergence is at least 2(-1)|m(3)(mu)|n(-1/2) +O(n(-1)), and when we have zero third moment, then the rate of convergence is 2(-1)(m(4)(mu) - 1)n(-1) + O(n(-3/2)). To achieve this, we improve a previous result concerning the position and weights of two atoms that eventually appear in the converging measure. The proofs only require elementary techniques of complex analysis.