Structural, continuity, and asymptotic properties of a branching particle system

We delineate a connection between the stochastic evolution of the cluster structure of a specific branching-diffusing particle system and a certain previously unknown structure-invariance property of a related class of distributions. Thus, we demonstrate that a Pólya-Aeppli sum of i.i.d.r.v.'s with a common zero-modified geometric distribution also follows a Pólya-Aeppli law. The consideration of these classes is motivated by and applied to studying subtle properties of this branching-diffusing particle system, which belongs to the domain of attraction of a continuous Dawson-Watanabe superprocess. We illustrate this structure-invariance property by considering the Athreya-Ney-type representation of the cluster structure of our particle system. Also, we apply this representation to prove the continuity in mean square of a related real-valued stochastic process. In contrast to other works in this field, we impose the condition that the initial random number of particles follows a Pólya-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. Our results extend some recent work of Vinogradov. Specifically, we resolve the issue of noninvariance of the initial field and manage to avoid related anomalies that arose in earlier studies. Also, we demonstrate that under natural additional assumptions, our particle system must have evolved from a scaled Poisson field starting at a specified time. In some sense, this result provides a partial justification for assuming that the system had originated at a certain time in the past from a Poisson field of particles. We demonstrate that the corresponding high-density limit of our branching-diffusing particle system inherits an analogous backward-evolution property. Several of our results illustrate a general convergence theorem of Jørgensen et al. to members of the power-variance family of distributions. Finally, combining a Poisson mixture representation for the branching particle system considered with certain sharp analytical methods gives us an explicit representation for the leading error term of the high-density approximation as a linear combination of related Bessel functions. This refines a theorem of Vinogradov on the rate of convergence.