Small positive values for supercritical branching processes in random environment

TL;DR

This paper studies the rare events where supercritical branching processes in random environments remain small over time, characterizing their asymptotic probabilities and the influence on the most recent common ancestor.

Contribution

It provides a detailed asymptotic analysis of small positive values in supercritical BPREs, including explicit rates for linear fractional cases and the impact on ancestral structure.

Findings

Exponential decay rates for small population probabilities are characterized.

Explicit results are obtained for linear fractional reproduction laws.

The regimes influence the asymptotic behavior of the most recent common ancestor.

Abstract

Branching Processes in Random Environment (BPREs) $(Z_n:n\geq0)$ are the generalization of Galton-Watson processes where in each generation the reproduction law is picked randomly in an i.i.d. manner. In the supercritical case, the process survives with positive probability and then almost surely grows geometrically. This paper focuses on rare events when the process takes positive but small values for large times. We describe the asymptotic behavior of $\mathbb{P}(1 \leq Z_n \leq k|Z_0=i)$, $k,i\in\mathbb{N}$ as $n\rightarrow \infty$. More precisely, we characterize the exponential decrease of $\mathbb{P}(Z_n=k|Z_0=i)$ using a spine representation due to Geiger. We then provide some bounds for this rate of decrease. If the reproduction laws are linear fractional, this rate becomes more explicit and two regimes appear. Moreover, we show that these regimes affect the asymptotic behavior of the most recent common ancestor, when the population is conditioned to be small but positive for large times.