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Minimum and extremal process for a branching random walk outside the boundary case

Xinxin Chen, Haojie Hou

TL;DR

This work analyzes the minimum and extremal process for a supercritical branching random walk outside the boundary case, where the spine displacements have a stretched-exponential left tail with exponent $b\in(0,1/2)$. By leveraging spine decomposition and a refined near-time large-jump mechanism, the authors prove weak convergence for the minimum and for the extremal process when centered by $\alpha_n=\lambda(mn)^b-a\log n$, and they establish an almost-sure infimum over infinite rays governed by a deterministic constant $a_*\in(0,\infty)$. The limiting objects are decorated Poisson-type processes driven by the almost-sure limit $W_\infty$ of the additive martingale, with an explicit limiting intensity and decorations derived from independent BRWs. The results confirm Barral–Hu–Madaule’s conjecture in the stretched-exponential regime with $b\in(0,1/2)$ and illuminate how non-boundary BRWs exhibit distinct extremal behavior from boundary cases, with potential implications for extreme-value theory in branching structures.

Abstract

This work extends the studies on the minimum and extremal process of a supercritical branching random walk outside the boundary case which cannot be reduced to the boundary case. We study here the situation where the log-generating function explodes at $1$ and the random walk associated to the spine possesses a stretched exponential tail with exponent $b\in(0,\frac12)$. Under suitable conditions, we confirm the conjecture of Barral, Hu and Madaule [Bernoulli 24(2) 2018 801-841], and obtain the weak convergence for the minimum and the extremal process. We also establish an a.s. infimum result over all infinity rays of this system.

Minimum and extremal process for a branching random walk outside the boundary case

TL;DR

This work analyzes the minimum and extremal process for a supercritical branching random walk outside the boundary case, where the spine displacements have a stretched-exponential left tail with exponent . By leveraging spine decomposition and a refined near-time large-jump mechanism, the authors prove weak convergence for the minimum and for the extremal process when centered by , and they establish an almost-sure infimum over infinite rays governed by a deterministic constant . The limiting objects are decorated Poisson-type processes driven by the almost-sure limit of the additive martingale, with an explicit limiting intensity and decorations derived from independent BRWs. The results confirm Barral–Hu–Madaule’s conjecture in the stretched-exponential regime with and illuminate how non-boundary BRWs exhibit distinct extremal behavior from boundary cases, with potential implications for extreme-value theory in branching structures.

Abstract

This work extends the studies on the minimum and extremal process of a supercritical branching random walk outside the boundary case which cannot be reduced to the boundary case. We study here the situation where the log-generating function explodes at and the random walk associated to the spine possesses a stretched exponential tail with exponent . Under suitable conditions, we confirm the conjecture of Barral, Hu and Madaule [Bernoulli 24(2) 2018 801-841], and obtain the weak convergence for the minimum and the extremal process. We also establish an a.s. infimum result over all infinity rays of this system.
Paper Structure (19 sections, 17 theorems, 248 equations)

This paper contains 19 sections, 17 theorems, 248 equations.

Table of Contents

  1. Introduction and main results
  2. Background introduction
  3. Boundary case (I).
  4. Outside boundary case: case (III).
  5. Outside boundary case: case (II)
  6. Main results
  7. Proof strategies and discussions
  8. Elementary properties for random walk $S_n$
  9. Proofs of the Theorems \ref{['thm1']} and \ref{['thm2']}
  10. Proof of Theorem \ref{['thm3']}
  11. Proof of Theorem \ref{['thm3']} for $a_*>0$.
  12. Proof of Theorem \ref{['thm3']} for $a_*<\infty$.
  13. Proof of Proposition \ref{['prop:key-proposition']}
  14. Step 1.
  15. Step 2.
  16. ...and 4 more sections

Key Result

Theorem 1.1

Assume Assumption1, Assumption3b and Assumption2 with $b\in (0, \frac{1}{2})$. Then for any $x\in \mathbb{R}$ and $f\in \mathcal{S}$, where $C^*(f_x)$ is given as in Def-of-C-star-f with $f_x:=f(\cdot+x)$. Moreover, for any non-negative function $f$ with bounded support, we have

Theorems & Definitions (18)

  • Theorem 1.1
  • Remark 1.2
  • Corollary 1.3
  • Theorem 1.4
  • Theorem 1.5
  • Lemma 2.1
  • Lemma 2.2
  • Lemma 2.3
  • Lemma 2.4
  • Lemma 2.5
  • ...and 8 more