Non-commutative law of rare events

Abstract

We establish quantitative versions of the law of rare events and binomial approximations in non-commutative probability settings, including the free, Boolean, and monotone convolution frameworks. Our main results provide explicit error bounds in the non-commutative Wasserstein distance for approximations of convolutions of rare countings by non-commutative Poisson and binomial distributions. These bounds extend classical results from the tensor setting to the non-commutative regime. Our approach relies on a discrete Lindeberg-type interpolation scheme combined with algebraic properties of cumulants adapted to each independence notion. The results presented here fill a gap in the literature concerning explicit rates of convergence in non-commutative limit theorems.