论文标题
分支随机步行的水平集的较低偏差概率
Lower deviation probabilities for level sets of the branching random walk
论文作者
论文摘要
给定一个随机步行$ \ {z_n \} _ {n \ geq0} $上的$ \ mathbb {r} $，让$ z_n（[y，\ infty））$是位于$ n $ n $ n $ n $ n $ n $ n $ n $ n $ n $ n $ [y，\ infty）中的粒子数量。从\ cite {biggins1977}中知道，在某些温和条件下，$ n^{ - 1} \ log z_n（[[θx^* n，\ infty））$ convery a.s.到$ \ log m-i（θx^*）$，其中$ \ log m-i（θx^*）$是一个正常数。在这项工作中，我们研究了其较低的偏差，换句话说，$ \ mathbb {p} \ left的收敛速率（z_n（z_n（[θx^* n，\ infty）））<e^{an} \ right），$ a \ in $ a \其中$ a \ in [0，0，\ log log log m-i（\ log m-i（fiol m-i（f）m-i mi（θx^**））$。我们的结果完成了\ cite {mehmet}，\ cite {helower}和\ cite {gwlower}中的结果。
Given a branching random walk$\{Z_n\}_{n\geq0}$ on $\mathbb{R}$, let $Z_n([y,\infty))$ be the number of particles located in $[y,\infty)$ at generation $n$. It is known from \cite{Biggins1977} that under some mild conditions, $n^{-1}\log Z_n([θx^* n,\infty))$ converges a.s. to $\log m-I(θx^*)$, where $\log m-I(θx^*)$ is a positive constant. In this work, we investigate its lower deviation, in other words, the convergence rates of $$\mathbb{P}\left(Z_n([θx^* n,\infty))<e^{an}\right),$$ where $a\in[0,\log m-I(θx^*))$. Our results complete those in \cite{Mehmet}, \cite{Helower} and \cite{GWlower}.
