Distributional comparison for non-commutative infinitely divisible probability measures

Abstract

We determine ``cumulant-type'' upper bounds of the non-commutative Wasserstein distance for certain classes of distributions $\hat{I}¼$ and $\hat{I}½$, which are infinite divisible with respect to the Boolean, classical and free convolutions. The main contribution of the manuscript is an estimation of the non-commutative Wasserstein distance between $\hat{I}¼$ and $\hat{I}½$, expressed in terms of the difference between cumulants of order less than $2m+4$.