标题： 关于具有重尾的分支随机游走经验分布的大偏差概率 显示英文标题

标题： On large deviation probabilities for empirical distribution of branching random walks with heavy tails

摘要： 给定一个分支随机游走$(Z_n)_{n\geq0}$在$\mathbb{R}$上，令$Z_n(A)$为在第$n$代位于区间$A$中的粒子数。 众所周知（例如，\cite{biggins}），在一些温和条件下，$Z_n(\sqrt nA)/Z_n(\mathbb{R})$几乎 surely 收敛到$\nu(A)$当$n\rightarrow\infty$，其中$\nu$是标准高斯测度。 在本工作中，我们研究在步长或后代分布具有重尾的情况下其大偏差概率，即 $$\mathbb{P}(Z_n(\sqrt nA)/Z_n(\mathbb{R})>p)$$的衰减率作为$n\rightarrow\infty$，其中$p\in(\nu(A),1)$。我们的结果补充了\cite{ChenHe}和\cite{Louidor}中的结果。 显示英文摘要

摘要： Given a branching random walk $(Z_n)_{n\geq0}$ on $\mathbb{R}$, let $Z_n(A)$ be the number of particles located in interval $A$ at generation $n$. It is well known (e.g., \cite{biggins}) that under some mild conditions, $Z_n(\sqrt nA)/Z_n(\mathbb{R})$ converges a.s. to $\nu(A)$ as $n\rightarrow\infty$, where $\nu$ is the standard Gaussian measure. In this work, we investigate its large deviation probabilities under the condition that the step size or offspring law has heavy tail, i.e. the decay rate of $$\mathbb{P}(Z_n(\sqrt nA)/Z_n(\mathbb{R})>p)$$ as $n\rightarrow\infty$, where $p\in(\nu(A),1)$. Our results complete those in \cite{ChenHe} and \cite{Louidor}.