Right-Most Position of a Last Progeny Modified Time Inhomogeneous Branching Random Walk

[ Joint work with Antar Bandyopadhyay ]

Abstract : In this work, we consider a modification of the time inhomogeneous branching random walk, where the driving increment distribution changes over time macroscopically. Following Bandyopadhyay and Ghosh (arXiv:2106.02880), we give certain independent and identically distributed (i.i.d.) displacements to all the particles at the last generation. We call this process last progeny modified time inhomogeneous branching random walk (LPMTI-BRW). Under very minimal assumptions on the underlying point processes of the displacements, we show that the maximum displacement converges to a non-trivial limit after an appropriate centering which is either linear or linear with a logarithmic correction. Interestingly, the limiting distribution depends only on the first set of increments. We also derive Brunet–Derrida-type results of point process convergence of our LPMTI-BRW to a decorated Poisson point process. As in the case of the maximum, the limiting point process also depends only on the first set of increments. Our proofs are based on a method of coupling the maximum displacement with an appropriate linear statistics, which was introduced by Bandyopadhyay and Ghosh (arXiv:2106.02880).

Journal : Statistics & Probability Letters, 193: Paper No. 109697 (2023)

DOI : 10.1016/j.spl.2022.109697

arXiv : 2110.04532