Constructions of random distributions via sequential barycenters

Abstract

This article introduces and develops a constructive method for generating random probability measures with a prescribed mean or distribution of the means. The method involves sequentially generating an array of barycenters which uniquely defines a probability measure. Basic properties of the generated measures are presented, including conditions under which almost all the generated measures are continuous or almost all are purely discrete or almost all have finite support. Applications are given to models for average-optimal control problems and to experimental approximation of universal constants.