Large deviations for the right-most position of a last progeny modified branching random walk

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 [2] 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 Höfelsauer [7].