Large Deviations for the Right-Most Position of a Last Progeny Modified Branching Random Walk

TL;DR

This paper establishes large deviation principles for the maximum particle position in a modified branching random walk where the last generation's displacements are independently altered, extending classical results.

Contribution

It derives the large deviation principle for the right-most position in a modified BRW and completes the LDP analysis for the classical model under minimal assumptions.

Findings

LDP for the right-most position in LPM-BRW established

Completes the LDP for classical BRW

Applicable under minimal assumptions

Abstract

In this work, we consider a modification of the usual Branching Random Walk (BRW), where we give certain independent and identically distributed (i.i.d.) displacements to all the particles at the $n$-th generation, which may be different from the driving increment distribution. This model was first introduced by Bandyopadhyay and Ghosh (2021) and they termed it as Last Progeny Modified Branching Random Walk (LPM-BRW). Under very minimal assumptions, we derive the large deviation principle (LDP) for the right-most position of a particle in generation $n$. As a byproduct, we also complete the LDP for the classical model, which complements the earlier work by Gantert and H\"{o}felsauer (2018).