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Extremal Process of Last Progeny Modified Branching Random Walks

Partha Pratim Ghosh, Bastien Mallein

TL;DR

This work analyzes the extremal process of a last-progeny modified branching random walk, where final-generation particle positions are perturbed by i.i.d. variables with an exponential tail. Depending on the tail parameter relative to the BRW boundary, the extremal process converges to distinct limiting objects: a decorated Poisson process in the subcritical regime, a derivative-martingale–weighted Poisson process at criticality, and a perturbed BRW edge in the supercritical regime. The analysis hinges on a Laplace-transform approach, the many-to-one lemma, and spine/decomposition techniques, connecting the perturbation tail to the BRW's extremal structure via additive and derivative martingales. The results extend the understanding of noisy BRWs by identifying precise centering and limiting point-process structures across regimes, with implications for the interplay between BRW geometry and heavy/light-tailed perturbations.

Abstract

We consider a last progeny modified branching random walk, in which the position of each particle at the last generation $n$ is modified by an i.i.d. copy of a random variable $Y$. Depending on the asymptotic properties of the tail of $Y$, we describe the asymptotic behaviour of the extremal process of this model as $n \to \infty$.

Extremal Process of Last Progeny Modified Branching Random Walks

TL;DR

This work analyzes the extremal process of a last-progeny modified branching random walk, where final-generation particle positions are perturbed by i.i.d. variables with an exponential tail. Depending on the tail parameter relative to the BRW boundary, the extremal process converges to distinct limiting objects: a decorated Poisson process in the subcritical regime, a derivative-martingale–weighted Poisson process at criticality, and a perturbed BRW edge in the supercritical regime. The analysis hinges on a Laplace-transform approach, the many-to-one lemma, and spine/decomposition techniques, connecting the perturbation tail to the BRW's extremal structure via additive and derivative martingales. The results extend the understanding of noisy BRWs by identifying precise centering and limiting point-process structures across regimes, with implications for the interplay between BRW geometry and heavy/light-tailed perturbations.

Abstract

We consider a last progeny modified branching random walk, in which the position of each particle at the last generation is modified by an i.i.d. copy of a random variable . Depending on the asymptotic properties of the tail of , we describe the asymptotic behaviour of the extremal process of this model as .
Paper Structure (11 sections, 9 theorems, 101 equations)

This paper contains 11 sections, 9 theorems, 101 equations.

Table of Contents

  1. Introduction
  2. Extremal process in the subcritical regime
  3. Extremal process in the critical regime
  4. Extremal process in the supercritical regime
  5. Heuristics for our results
  6. Organization of the article.
  7. Laplace transform of the last progeny modified branching random walk
  8. Proof of the theorems
  9. BRW below the boundary case
  10. BRW in the boundary case
  11. BRW above the boundary case

Key Result

Theorem 1.2

Let $\theta > 0$ such that $\kappa(\theta) < \infty$. We assume that $(W_n(\theta), n \geq 0)$ is uniformly integrable. Let $\nu$ be a probability distribution on $\mathbb{R}$ such that there exists a constant $L\in (0,\infty)$ satisfying Then, writing the extremal process $\tau_{-m_n} \mathcal{E}_n$ converges in law to a PPP$(\theta W_\infty(\theta)\mathrm{e}^{-\theta x} \,\mathrm{d} x)$.

Theorems & Definitions (12)

  • Remark 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Remark 1.5
  • Remark 1.6
  • Theorem 1.7
  • Lemma 2.1
  • Lemma 2.2
  • Lemma 2.3
  • ...and 2 more