Tightness for Maxima of Generalized Branching Random Walks

TL;DR

This paper investigates the maxima of generalized branching random walks with dependencies, proving tightness of their distributions after median adjustment under certain tail conditions, and demonstrating exponential decay of their right tails.

Contribution

It establishes tightness and tail decay properties for maxima of generalized branching random walks with dependencies, extending previous results to more complex models.

Findings

Proves tightness of maxima distributions after median shift.

Shows exponential decay of the right tail of maxima distributions.

Extends classical results to models with time and local dependence.

Abstract

We study generalized branching random walks, which allow time dependence and local dependence between siblings. Under appropriate tail assumptions, we prove the tightness of $F_n(\cdot-Med(F_n))$, where $F_n(\cdot)$ is the maxima distribution at time $n$ and $Med(F_n)$ is the median of $F_n(\cdot)$. The main component in the argument is a proof of exponential decay of the right tail $1-F_n(\cdot-Med(F_n))$.