Colloquium

ABSTRACT: We delineate a connection between the stochastic evolution of the cluster structure of a branching-diffusing particle system that belongs to the domain of attraction of a continuous Dawson-Watanabe superprocess and a certain previously unknown structure-invariance property of a related class of probability distributions. We illustrate the structure invariance by considering the Athreya-Ney-type representation of the cluster structure of the particle system, and we apply this representation to prove the continuity in mean square of a related real-valued stochastic process. Finally, by combining a Poisson mixture representation for the branching particle system with certain sharp analytical methods, we get an explicit representation for the leading error term of the high-density approximation as a linear combination of related Bessel functions. In contrast to other works in this field, we impose the condition that the initial random number of particles follows a P\'{o}lya-Aeppli law, a condition that is consistent with stochastic models that emerge in such varied fields as population genetics, ecology, insurance risk, and bacteriophage growth.