A large-deviation principle for the empirical distribution of a regular branching random walk
Abstract
Let {Zn}n≥0 be a supercritical branching random walk with deterministic rooted b-ary tree, b≥2, and symmetric displacements satisfying limx→+∞x−αlogPp(X>x)=−λ with α,λ>0. Set Z̅n(·):=Zn(·)/Zn(ℝ), n≥0. For a finite union A of intervals, we establish the following full large-deviation principle: for every Borel set Γ⊂[0,1], −infq∈Γ∘QA(q) &≤ liminfn→∞n−α/2 logPp (Z̅n(√n A)∈Γ) &≤ limsupn→∞n−α/2 logPp (Z̅n(√n A)∈Γ) &≤ −infq∈Γ̅QA(q), where QA is a good rate function on [0,1]. Meanwhile, we obtain large deviation probabilities for {Z̅n(√n A)}n≥1. This strengthens the nonmatching upper and lower bounds obtained by Chen and He [Probab. Theory Related Fields 175 (2019) 255-307] for regular trees with Weibull displacements. Our method combines a large-deviation principle for rescaled displacement tree fields and exponential equivalence.
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BibTeX
@misc{zhang2026largedeviation,
title = {{A large-deviation principle for the empirical distribution of a regular branching random walk}},
author = {Shuxiong Zhang and Yaping Zhu},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.08714},
eprint = {2609.08714},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.08714},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - A large-deviation principle for the empirical distribution of a regular branching random walk
AU - Shuxiong Zhang
AU - Yaping Zhu
PY - 2026
DA - 2026/09/08
PB - arXiv
DO - 10.48550/arxiv.2609.08714
UR - https://arxiv.org/abs/2609.08714
AB - Let {Z_n}_n≥0 be a supercritical branching random walk with deterministic rooted b-ary tree, b≥2, and symmetric displacements satisfying lim_x→+∞x^−αlogPp(X>x)=−λ with α,λ>0. Set Z̅_n(·):=Z_n(·)/Z_n(ℝ), n≥0. For a finite union A of intervals, we establish the following full large-deviation principle: for every Borel set Γ⊂[0,1], −inf_q∈Γ^∘Q_A(q) &≤ liminf_n→∞n^−α/2 logPp (Z̅_n(√n A)∈Γ) &≤ limsup_n→∞n^−α/2 logPp (Z̅_n(√n A)∈Γ) &≤ −inf_q∈Γ̅Q_A(q), where Q_A is a good rate function on [0,1]. Meanwhile, we obtain large deviation probabilities for {Z̅_n(√n A)}_n≥1. This strengthens the nonmatching upper and lower bounds obtained by Chen and He [Probab. Theory Related Fields 175 (2019) 255-307] for regular trees with Weibull displacements. Our method combines a large-deviation principle for rescaled displacement tree fields and exponential equivalence.
ER -