Functional limit theorem for branching processes in nearly degenerate varying environment

TL;DR

This paper establishes functional limit theorems for branching processes in nearly degenerate environments, revealing the asymptotic behavior conditioned on non-extinction or with immigration, extending previous one-dimensional results.

Contribution

It introduces new functional limit theorems for branching processes in nearly degenerate environments, considering conditioning and immigration, expanding the understanding of their long-term behavior.

Findings

Limiting process is a time-changed birth-death process conditioned on survival.

Another limiting process is a time-changed stationary branching process with immigration.

Results extend previous one-dimensional limit theorems to functional convergence.

Abstract

We investigate branching processes in nearly degenerate varying environment, where the offspring distribution converges to the degenerate distribution at 1. Such processes die out almost surely, therefore, we condition on non-extinction or add inhomogeneous immigration. Extending our one-dimensional limit results we derive functional limit theorems. In the former case, the limiting process is a time-changed simple birth-and-death process on $(-\infty, 0]$ conditioned on survival at $0$, while in the latter, it is a time-changed stationary continuous time branching process with immigration.