Branching random walk conditioned on large martingale limit

TL;DR

This paper analyzes the behavior of a branching random walk conditioned on extreme values of its martingale limit, revealing joint tail distributions and convergence properties in atypical scenarios.

Contribution

It establishes the joint tail distribution of the martingale limit and the global minimum, and studies the process conditioned on these rare events.

Findings

Joint tail distribution of $W_inite$ and the global minimum

Convergence in law of the process conditioned on atypical events

Right tail of the derivative martingale limit

Abstract

We consider a branching random walk in the non-boundary case where the additive martingale $W_n$ converges a.s. and in mean to some non-degenerate limit $W_\infty$. We first establish the joint tail distribution of $W_\infty$ and the global minimum of this branching random walk. Next, conditioned on the event that the minimum is atypically small or conditioned on very large $W_\infty$, we study the branching random walk viewed from the minimum and obtain the convergence in law in the vague sense. As a byproduct, we also get the right tail of the limit of derivative martingale.