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

• Main Authors: Bandyopadhyay, Antar, Ghosh, Partha Pratim
• Format: Journal Article
• Language: English
• Published: 09-10-2021
• Subjects: Mathematics - Probability
• DOI: 10.48550/arxiv.2110.04532

Statistics & Probability Letters, 193: Paper No. 109697 (2023),
1-8 In this work, we consider a modification of time \emph{inhomogeneous}
branching random walk, where the driving increment distribution changes over
time macroscopically. Following Bandyopadhyay and Ghosh (2021), we give certain
independent and identically distributed (i.i.d.) displacements to all the
particles at the last generation. We call this process \emph{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 limit after only 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 the method of coupling the maximum
displacement with the smoothing transformation, which was introduced by
Bandyopadhyay and Ghosh (2021).