论文标题
关于大尾巴的分支随机步行的经验分布的大偏差概率
On large deviation probabilities for empirical distribution of branching random walks with heavy tails
论文作者
论文摘要
给定$ \ mathbb {r} $上的分支随机步行$（z_n）_ {n \ geq0} $，让$ z_n（a）$是$ n $ n Interval $ a $ a $ a $ a $ n $的粒子的数量。众所周知（例如，\ cite {biggins}），在某些温和条件下，$ z_n（\ sqrt na）/z_n（\ mathbb {r}）$ commentes a.s.至$ν（a）$ as $ n \ rightarrow \ infty $，其中$ν$是标准高斯度量。在这项工作中，我们研究了其较大的偏差概率，条件是步骤大小或后代定律具有沉重的尾巴，即$$ \ mathbb {p}（z_n（\ sqrt na）/z_n（\ sqrt na）/z_n（\ mathbb {r}）> p）$ n as $ n \ firtarrow $ p p（n $ p p）我们的结果完成了\ cite {chenhe}和\ cite {louidor}中的结果。
Given a branching random walk $(Z_n)_{n\geq0}$ on $\mathbb{R}$, let $Z_n(A)$ be the number of particles located in interval $A$ at generation $n$. It is well known (e.g., \cite{biggins}) that under some mild conditions, $Z_n(\sqrt nA)/Z_n(\mathbb{R})$ converges a.s. to $ν(A)$ as $n\rightarrow\infty$, where $ν$ is the standard Gaussian measure. In this work, we investigate its large deviation probabilities under the condition that the step size or offspring law has heavy tail, i.e. the decay rate of $$\mathbb{P}(Z_n(\sqrt nA)/Z_n(\mathbb{R})>p)$$ as $n\rightarrow\infty$, where $p\in(ν(A),1)$. Our results complete those in \cite{ChenHe} and \cite{Louidor}.
