arXiv 1807.082632018-07-22math.PR

分支随机游动最大值的下偏差和中偏差概率

Lower deviation and moderate deviation probabilities for maximum of a branching random walk

Xinxin Chen, Hui He

备注：: 33 pages, 2 figures

摘要：

给定$\mathbb R$上从原点开始的超临界分支随机游动，令$M_n$为第$n$代个体的最大位置。在适当的条件下，由\cite{A13}可知，对于适当的常数$x^*$和$θ^*$，$M_n-x^*n+\frac{3}{2θ^*}\log n$按规律收敛于$n\rightarrow\infty$.在这项工作中，我们研究其适度的偏差，换句话说，$$\mathbb{P}\left(M_n\leq x^*n-\frac{3}{2θ^*}\log n-\ell_n\right),$$的收敛速度的任何正序列$(\ell_n)$，使$\ell_n=O(n)$和$\ell_n\uparrow\infty$。作为副产品，我们还获得了PDMATHAMN的较低偏差;即，在Böttcher情形下，当后代数至少为2时，$x<x^*$的\[ \mathbb{P}(M_n\leq xn), \]的收敛速度。最后，我们应用我们的技巧研究了微分鞅极限的小球概率。

英文摘要：

Given a super-critical branching random walk on $\mathbb R$ started from the origin, let $M_n$ be the maximal position of individuals at the $n$-th generation. Under some mild conditions, it is known from \cite{A13} that as $n\rightarrow\infty$, $M_n-x^*n+\frac{3}{2θ^*}\log n$ converges in law for some suitable constants $x^*$ and $θ^*$. In this work, we investigate its moderate deviation, in other words, the convergence rates of $$\mathbb{P}\left(M_n\leq x^*n-\frac{3}{2θ^*}\log n-\ell_n\right),$$ for any positive sequence $(\ell_n)$ such that $\ell_n=O(n)$ and $\ell_n\uparrow\infty$. As a by-product, we also obtain lower deviation of $M_n$; i.e., the convergence rate of \[ \mathbb{P}(M_n\leq xn), \] for $x<x^*$ in Böttcher case where the offspring number is at least two. Finally, we apply our techniques to study the small ball probability of limit of derivative martingale.

机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。

摘要：

英文摘要：

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