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Precise upper deviation estimates for the maximum of a branching random walk

Lianghui Luo

TL;DR

This work derives precise upper large deviation asymptotics for the maximal displacement $M_n$ in a branching random walk and describes the extremal process under the conditioning event $\\{M_n \\ge n\psi'(\\theta) \\}$, introducing a limiting decoration point process $D(\\theta)$ that governs the limit law. The analysis combines spine decomposition, a time-reversal construction of an auxiliary process $D_n^{\\theta}$, and a non-lattice local limit theorem to obtain sharp tail estimates and to characterize the joint limit of the extremal process with the centered maximum. The main results include an explicit tail constant $C(\\theta)$ and a conditional weak convergence $(\\mathcal{E}_n, M_n-n\\psi'(\\theta)) \\xrightarrow{d} (D(\\theta), \\mathbf{e})$ where $\\mathbf{e}$ is Exp$(\\theta)$, as well as a detailed description of when $D(\\theta)$ is finite or infinite. These findings generalize and extend prior work on branching Brownian motion to discrete BRWs under non-lattice and moment conditions, providing a refined understanding of the fine structure of extremes and their decorations.

Abstract

We consider the precise upper large deviations estimates for the maximal displacement of a branching random walk. In addition, we obtain a description of the extremal process of the branching random walk conditioned on this large deviations event. This introduces a family of point measure playing a role similar to the decoration measures introduced in [9] for branching Brownian motion.

Precise upper deviation estimates for the maximum of a branching random walk

TL;DR

This work derives precise upper large deviation asymptotics for the maximal displacement in a branching random walk and describes the extremal process under the conditioning event , introducing a limiting decoration point process that governs the limit law. The analysis combines spine decomposition, a time-reversal construction of an auxiliary process , and a non-lattice local limit theorem to obtain sharp tail estimates and to characterize the joint limit of the extremal process with the centered maximum. The main results include an explicit tail constant and a conditional weak convergence where is Exp, as well as a detailed description of when is finite or infinite. These findings generalize and extend prior work on branching Brownian motion to discrete BRWs under non-lattice and moment conditions, providing a refined understanding of the fine structure of extremes and their decorations.

Abstract

We consider the precise upper large deviations estimates for the maximal displacement of a branching random walk. In addition, we obtain a description of the extremal process of the branching random walk conditioned on this large deviations event. This introduces a family of point measure playing a role similar to the decoration measures introduced in [9] for branching Brownian motion.
Paper Structure (13 sections, 20 theorems, 210 equations)

This paper contains 13 sections, 20 theorems, 210 equations.

Table of Contents

  1. Introduction
  2. Spine decomposition from the tip of the branching random walk
  3. Ulam-Harris-Neveu notation
  4. Spine decomposition: a change of measure
  5. Applications of local limit theorem of a non-lattice random walk
  6. Limit estimates for the auxiliary point process
  7. Proof of main results
  8. Proof of Theorems \ref{['th2']} and \ref{['th1']}
  9. Proof of Proposition \ref{['log']}
  10. Some extra properties of the extremal process of conditioned branching random walk
  11. Subcritical branching random walk
  12. Extremal process as a conditioned point measure
  13. Continuity of the extremal process

Key Result

Proposition 1.1

Assume that there exists some $\theta>0$ satisfying $\psi(\theta)<\infty$. If $\psi(0)>0$, we have If $\psi(0)\leq 0$, we have

Theorems & Definitions (42)

  • Proposition 1.1
  • Remark 1.1
  • Theorem 1.2
  • Remark 1.2
  • Theorem 1.3
  • Remark 1.3
  • Lemma 2.1: Theorem 1.1 inshi
  • Theorem 2.2: Lyonslyons
  • Lemma 2.3
  • proof
  • ...and 32 more