Upper large deviations for Branching Processes in Random Environment with heavy tails

TL;DR

This paper analyzes the upper large deviations of Branching Processes in Random Environments with heavy tails, providing explicit rate functions and phase transition descriptions, generalizing previous results for non-heavy-tailed cases.

Contribution

It derives the upper large deviation rate function for BPREs with heavy tails, linking it to the environment's random walk and tail decay, and extends known results to broader tail conditions.

Findings

Explicit expression for the upper rate function of BPREs with heavy tails

Interpretation of the rate function in terms of least costly growth paths

Description of phase transitions in the large deviation behavior

Abstract

Branching Processes in a Random Environment (BPREs) $(Z_n:n\geq0)$ are a generalization of Galton Watson processes where in each generation the reproduction law is picked randomly in an i.i.d. manner. We determine here the upper large deviation of the process when the reproduction law may have heavy tails. The behavior of BPREs is related to the associated random walk of the environment, whose increments are distributed like the logarithmic mean of the offspring distributions. We obtain an expression of the upper rate function of $(Z_n:n\geq0)$, that is the limit of $-\log \mathbb{P}(Z_n\geq e^{\theta n})/n$ when $n\to \infty$. It depends on the rate function of the associated random walk of the environment, the logarithmic cost of survival $\gamma:=-\lim_{n\to\infty} \log \mathbb{P}(Z_n>0)/n$ and the polynomial decay $\beta$ of the tail distribution of $Z_1$. We give interpretations of this rate function in terms of the least costly ways for the process $(Z_n: n \geq 0)$ of attaining extraordinarily large values and describe the phase transitions. We derive then the rate function when the reproduction law does not have heavy tails, which generalizes the results of B\"oinghoff and Kersting (2009) and Bansaye and Berestycki (2008) for upper large deviations. Finally, we specify the upper large deviations for the Galton Watson processes with heavy tails.