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 (arXiv:2106.02880) 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 (Electron. Commun. Probab. 23 (2018), no. 34, 1–12).

Journal : Electronic Communications in Probability, 27: Paper No. 6 (2022)

DOI : 10.1214/22-ECP446

arXiv : 2106.08093