Branching random walk conditioned on large martingale limit

TL;DR

This work analyzes a branching random walk in the non-boundary regime by examining both the additive martingale $W_n$ and the derivative martingale $D_n$ and their almost sure and $L^p$ limits. Using Lyons' change of measure and spinal decomposition, it derives joint tail bounds for the martingale limit $W_\\infty$ and the global minimum $\\mathbf{M}$, and establishes asymptotic behavior for the tails of $D_\\infty$. The paper then proves tightness and vague convergence for the BRW viewed from its minimum under conditioning on extreme minimum values, revealing a limit that includes a Gaussian component and a random scale by the exponential of the minimum. It also links the large-$W_\\infty$ regime to the right tail of the derivative martingale and provides conditional convergence results when conditioning on large $W_\\infty$, with explicit tail constants and limiting distributions. Overall, the work connects extremal BRW behavior, martingale tails, and spine-based decompositions to describe the conditioned BRW dynamics and tail asymptotics in a coherent asymptotic framework.

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.

Figures (1)

• Figure 1: Time-reversing tree structure

Theorems & Definitions (45)

• Theorem 1.1
• Theorem 1.2
• Remark 1
• Remark 2
• Theorem 1.3
• Theorem 1.4
• Remark 3
• Proposition 2.1
• Proposition 2.2
• Lemma 2.3: Many-to-One