This website requires JavaScript.

Large deviation probabilities for the range of a d-dimensional supercritical branching random walk

Shuxiong Zhang
Dec 2022
摘要
Let $\{Z_n\}_{n\geq 0 }$ be a $d$-dimensional supercritical branching randomwalk started from the origin. Write $Z_n(S)$ for the number of particleslocated in a set $S\subset\mathbb{R}^d$ at time $n$. Denote by$R_n:=\inf\{\rho:Z_i(\{|x|\geq \rho\})=0,\forall~0\leq i\leq n\}$ the range of$\{Z_n\}_{n\geq 0 }$ before time $n$. In this work, we show that under somemild conditions $R_n/n$ converges in probability to some positive constant$x^*$ as $n\to\infty$. Furthermore, we study its corresponding lower and upperdeviation probabilities, i.e. the decay rates of $$ \mathbb{P}(R_n\leqxn)~\text{for}~x\in(0,x^*);~\mathbb{P}(R_n\geq xn) ~\text{for}~x\in(x^*,\infty)$$ as $n\to\infty$. As a by-product, we confirm a conjecture ofEngl\"{a}nder \cite{Englander04}.
展开全部
图表提取

暂无人提供速读十问回答

论文十问由沈向洋博士提出,鼓励大家带着这十个问题去阅读论文,用有用的信息构建认知模型。写出自己的十问回答,还有机会在当前页面展示哦。

Q1论文试图解决什么问题?
Q2这是否是一个新的问题?
Q3这篇文章要验证一个什么科学假设?
0
被引用
笔记
问答