# Large deviations for i.i.d. replications of the total progeny of a Galton--Watson process

## TL;DR

This paper explores large deviation principles for the total progeny in Galton--Watson processes, including cases with random initial populations and estimators of offspring mean, linking branching process theory with large deviation techniques.

## Contribution

It introduces large deviation results for total progeny distributions in Galton--Watson processes, including new insights for random initial populations and estimator sequences.

## Key findings

- Large deviation rate functions for total progeny are characterized.
- Results extend to processes with random initial populations.
- Estimates of offspring mean exhibit specific large deviation behaviors.

## Abstract

The Galton--Watson process is the simplest example of a branching process. The relationship between the offspring distribution, and, when the extinction occurs almost surely, the distribution of the total progeny is well known. In this paper, we illustrate the relationship between these two distributions when we consider the large deviation rate function (provided by Cram\'{e}r's theorem) for empirical means of i.i.d. random variables. We also consider the case with a random initial population. In the final part, we present large deviation results for sequences of estimators of the offspring mean based on i.i.d. replications of total progeny.