Weak Convergence Results for Multiple Generations of a Branching Process

TL;DR

This paper develops weak convergence limit theorems for multiple generations of critical and supercritical branching processes, leading to new functional CLTs, extremal results, and applications to ratios and maximums of generations.

Contribution

It introduces novel weak convergence results for joint behaviors of multiple generations, including an infinite-dimensional Brownian motion limit, extending classical branching process theory.

Findings

Functional CLT with infinite-dimensional Brownian motion limit

Darling-Erdős type extremal process results

CLT for ratios and maximums of generations

Abstract

We establish limit theorems involving weak convergence of multiple generations of critical and supercritical branching processes. These results arise naturally when dealing with the joint asymptotic behavior of functionals defined in terms of several generations of such processes. Applications of our main result include a functional central limit theorem (CLT), a Darling-Erd\"os result, and an extremal process result. The limiting process for our functional CLT is an infinite dimensional Brownian motion with sample paths in the infinite product space $(C_0[0,1])^{\infty}$, with the product topology, or in Banach subspaces of $(C_0[0,1])^{\infty}$ determined by norms related to the distribution of the population size of the branching process. As an application of this CLT we obtain a central limit theorem for ratios of weighted sums of generations of a branching processes, and also to various maximums of these generations. The Darling-Erd\"os result and the application to extremal distributions also include infinite dimensional limit laws. Some branching process examples where the CLT fails are also included.