Large Deviations for the Empirical Distribution in the General Branching Random Walk

TL;DR

This paper investigates the large deviations of the empirical distribution in a general branching random walk, revealing doubly exponential decay rates depending on the set and deviation magnitude.

Contribution

It provides the first detailed analysis of large deviation probabilities for empirical distributions in minimal assumption branching random walks.

Findings

Decay rates are doubly exponential in time.

Decay speed is either linear or square root of generations.

Explicit rate functions are derived for different deviations.

Abstract

We consider the general branching random walk under minimal assumptions, which in particular guarantee that the empirical particle distribution admits an almost sure central limit theorem. For such a process, we study the large time decay of the probability that the fraction of particles in a typical set deviates from its typical value. We show that such probabilities decay doubly exponentially with speed which is either linear in or the square root of the number of generations, depending on the set and the magnitude of the deviation. We also find the rate of decay in each case.