论文标题
D维超临界分支随机步行范围的较大偏差概率
Large deviation probabilities for the range of a d-dimensional supercritical branching random walk
论文作者
论文摘要
令$ \ {z_n \} _ {n \ geq 0} $为$ d $ - 维度超临界分支随机步行从原点开始。为位于$ s \ subset \ mathbb {r}^d $ in Time $ n $中的粒子数字编写$ z_n（s）$。用$ r_n：= \ inf \ {ρ：z_i（\ {| x | x | \ geqρ\}）= 0，\ forall〜0 \ 0 \ leq i \ leq n \} $ $ \ {z_n \}在这项工作中，我们表明，在某些温和条件下，$ r_n/n $的概率收敛于某些正常数$ x^*$，为$ n \ to \ infty $。此外，我们研究了其相应的下偏差概率和相应的上部偏差概率，即$$ \ mathbb {p}（r_n \ leq xn）〜\ text {for} 〜x \ in（0，x^*）; x \ in（x^*，\ infty）$$ as $ n \ to \ infty $。作为副产品，我们确认了Engländer\ Cite {Englander04}的猜想。
Let $\{Z_n\}_{n\geq 0 }$ be a $d$-dimensional supercritical branching random walk started from the origin. Write $Z_n(S)$ for the number of particles located in a set $S\subset\mathbb{R}^d$ at time $n$. Denote by $R_n:=\inf\{ρ:Z_i(\{|x|\geq ρ\})=0,\forall~0\leq i\leq n\}$ the range of $\{Z_n\}_{n\geq 0 }$ before time $n$. In this work, we show that under some mild conditions $R_n/n$ converges in probability to some positive constant $x^*$ as $n\to\infty$. Furthermore, we study its corresponding lower and upper deviation probabilities, i.e. the decay rates of $$ \mathbb{P}(R_n\leq xn)~\text{for}~x\in(0,x^*);~\mathbb{P}(R_n\geq xn) ~\text{for}~ x\in(x^*,\infty)$$ as $n\to\infty$. As a by-product, we confirm a conjecture of Engländer \cite{Englander04}.
