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 $$n=1,2,\ldots $$ n = 1 , 2 , … , a random time $$U_n$$ U n between 1 and n is chosen uniformly, and if the walker moved forward [resp. remained at rest] at time $$U_n$$ U n , then at time $$n+1$$ n + 1 it can move forward with probability p [resp. q ], or with probability $$1-p$$ 1 - p [resp. $$1-q$$ 1 - q ] it remains at its present position. For the case $$q>0$$ q > 0 , several limit theorems are obtained by Coletti, Gava, and de Lima (2019). In this paper we prove limit theorems for the case $$q=0$$ 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.

中文翻译：

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大象型“最懒惰”最小随机游走模型的极限定理

我们考虑由 Harbola、Kumar 和 Lindenberg（2014 年）引入的具有步进强化的一维离散时间随机游走的最小模型：步行者可以向前移动（永远不会向后），或保持静止。对于每个 $$n=1,2,\ldots $$n = 1 , 2 , ... ，统一选择 1 和 n 之间的随机时间 $$U_n$$ U n，如果步行者向前移动 [resp. 保持静止] 在时间 $$U_n$$ U n ，然后在时间 $$n+1$$ n + 1 它可以以概率 p [resp. q ]，或概率为 $$1-p$$ 1 - p [resp. $$1-q$$ 1 - q ] 它保持在当前位置。对于 $$q>0$$ q > 0 的情况，Coletti、Gava 和 de Lima (2019) 获得了几个极限定理。在本文中，我们证明了 $$q=0$$ q = 0 情况下的极限定理，其中随着 p 的变化，步行者可以表现出所有三种形式的渐近行为。作为副产品，