Comparing sums of exchangeable Bernoulli random variables

The paper is first concerned with a comparison of the partial sums associated with two sequences of n exchangeable Bernoulli random variables. it then considers a situation where such partial sums are obtained through an iterative procedure of branching type stopped at the first-passage time in a linearly decreasing upper barrier. These comparison results are illustrated with applications to certain urn models, sampling schemes and epidemic processes. A key tool is a non-standard hierarchical class of stochastic orderings between discrete random variables valued in {0, 1, ..., n}.