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Boundedness of discounted branching random walks via generic chaining

Zhenyuan Zhang

Abstract

Consider a discrete-time supercritical discounted branching random walk, in which increments at depth $k$ are independent and identically distributed with the same law as $m^{-kH}Y$, where $Y$ has a fixed law, $H>0$, and $m>1$ is the expected number of offspring at depth one. We provide a clean characterization of the boundedness of the discounted branching random walk: under mild conditions on the offspring distribution, the process is almost surely bounded if and only if $\mathbb{E}[|Y|^{1/H}]<\infty$. This extends results of Athreya (1985) and Aïdékon--Hu--Shi (2024), and provides a partial answer to Open Problem 31 of Aldous--Bandyopadhyay (2005).

Boundedness of discounted branching random walks via generic chaining

Abstract

Consider a discrete-time supercritical discounted branching random walk, in which increments at depth are independent and identically distributed with the same law as , where has a fixed law, , and is the expected number of offspring at depth one. We provide a clean characterization of the boundedness of the discounted branching random walk: under mild conditions on the offspring distribution, the process is almost surely bounded if and only if . This extends results of Athreya (1985) and Aïdékon--Hu--Shi (2024), and provides a partial answer to Open Problem 31 of Aldous--Bandyopadhyay (2005).
Paper Structure (11 sections, 12 theorems, 72 equations, 1 figure)

This paper contains 11 sections, 12 theorems, 72 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Sufficient condition for boundedness in the symmetric case
  3. Elementary lemmas
  4. Preliminaries and setup
  5. Constructing admissible sequences
  6. Removing bad events
  7. Proof of Theorem \ref{['thm']}
  8. Finiteness of the expected supremum
  9. Proof of the general asymmetric case
  10. Sufficiency for boundedness in the positive case
  11. Proofs of the main results

Key Result

Theorem 1

Assume that $m>1$ and $H>0$. The following statements hold. In particular, if $Z$ has finite moments of all orders and $\mathbb{P}(Z=0)=0$, then for all $H>0$, $\sup_{t\in\mathcal{T}}|X_t|<\infty$ a.s. if and only if $\mathbb{E}[|Y|^{1/H}]<\infty$.

Figures (1)

  • Figure 1: Illustrating the construction of the admissible sequence. Assume that the Galton--Watson tree is binary. Suppose that $u>0$ is some threshold, $h=2$, and we use solid dots to represent vertices $v$ such that $|m^{-H\ell(v)}\eta_v|> u$, where we recall that $\ell(v)$ is the depth of $v$. Assume also that all such vertices have depth at most 4. Our construction ensures that two rays belong to the same set in the partition only if they have a common ancestor at depth $h$ and the deepest solid dots on them coincide.

Theorems & Definitions (25)

  • Theorem 1
  • Remark 1
  • Proposition 2
  • Theorem 5
  • Lemma 6
  • proof
  • Lemma 7
  • proof
  • Lemma 8
  • proof
  • ...and 15 more