论文标题
限制“最懒惰”的最小随机步行模型的定理大象类型
Limit theorems for the 'laziest' minimal random walk model of elephant type
论文作者
论文摘要
我们考虑了由Harbola，Kumar和Lindenberg引入的一维离散时间随机步行的最小模型（2014年）：Walker可以前进（永不向后）或保持静止。对于每个$ n = 1,2，\ cdots $，一个随机的时间$ u_n $在$ 1 $和$ n $之间被统一选择，如果沃克向前移动[resp。在时间$ u_n $，然后在时间$ n+1 $时，它可以通过概率$ p $ [resp。 $ q $]，或带有概率$ 1-p $的[resp。 $ 1-q $]它仍然处于目前的位置。对于$ q> 0 $的情况，Coletti，Gava和de Lima（2019）获得了几个限制定理。在本文中，我们证明了$ q = 0 $的情况限制定理，其中沃克可以表现出所有三种形式的渐近行为，因为$ p $是多种多样的。作为副产品，我们在均匀的随机递归树上渗透中的根大小的簇大小获得了限制定理。
We consider a minimal model of one-dimensional discrete-time random walk with step-reinforcement, introduced by Harbola, Kumar, and Lindenberg (2014): The walker can move forward (never backward), or remain at rest. For each $n=1,2,\cdots$, a random time $U_n$ between $1$ and $n$ is chosen uniformly, and if the walker moved forward [resp. remained at rest] at time $U_n$, then at time $n+1$ it can move forward with probability $p$ [resp. $q$], or with probability $1-p$ [resp. $1-q$] it remains at its present position. For the case $q>0$, several limit theorems are obtained by Coletti, Gava, and de Lima (2019). In this paper we prove limit theorems for the case $q=0$, where the walker can exhibit all three forms of asymptotic behavior as $p$ is varied. As a byproduct, we obtain limit theorems for the cluster size of the root in percolation on uniform random recursive trees.
