标题： 对称树上简单随机游走的局部时间极值 显示英文标题

标题： Extremes of local times for simple random walks on symmetric trees

摘要： 我们考虑深度为$n$的$b$叉树上的简单随机游走的局部时间，并研究一个点过程，该过程编码具有最大局部时间的顶点的位置以及每个以$(n-r_n)$层为根的深度为$r_n$的子树的叶子上适当中心化的最大值，其中$(r_n)_{n \geq 1}$满足$\lim_{n \to \infty} r_n = \infty$和$\lim_{n \to \infty} r_n/n \in [0, 1)$。 我们证明点过程弱收敛到一个具有强度测度$\alpha Z_{\infty} (dx) \otimes e^{-2\sqrt{\log b}~y}dy$的 Cox 过程，其中$\alpha > 0$是一个常数，$Z_{\infty}$是在$[0, 1]$上的一个随机测度，其分布与临界随机乘法级联测度在尺度因子下的极限相同。 作为推论，我们建立了叶子上局部时间最大值的分布收敛到一个随机位移的 Gumbel 分布。 显示英文摘要

摘要： We consider local times of the simple random walk on the $b$-ary tree of depth $n$ and study a point process which encodes the location of the vertex with the maximal local time and the properly centered maximum over leaves of each subtree of depth $r_n$ rooted at the $(n-r_n)$ level, where $(r_n)_{n \geq 1}$ satisfies $\lim_{n \to \infty} r_n = \infty$ and $\lim_{n \to \infty} r_n/n \in [0, 1)$. We show that the point process weakly converges to a Cox process with intensity measure $\alpha Z_{\infty} (dx) \otimes e^{-2\sqrt{\log b}~y}dy$, where $\alpha > 0$ is a constant and $Z_{\infty}$ is a random measure on $[0, 1]$ which has the same law as the limit of a critical random multiplicative cascade measure up to a scale factor. As a corollary, we establish convergence in law of the maximum of local times over leaves to a randomly shifted Gumbel distribution.