Constructions of random distributions via sequential barycenters

This article introduces and develops a constructive method for generating r andom probability measures with a prescribed mean or distribution of the me ans. The method involves sequentially generating an array of barycenters wh ich uniquely defines a probability measure. Basic properties of the generat ed measures are presented, including conditions under which almost all the generated measures are continuous or almost all are purely discrete or almo st all have finite support. Applications are given to models for average-op timal control problems and to experimental approximation of universal const ants.