Let $$X_1, X_2, \ldots $$ be i.i.d. copies of some real random variable X. For any deterministic $$\varepsilon _2, \varepsilon _3, \ldots $$ in $$\{0,1\}$$ , a basic algorithm introduced by H.A. Simon yields a reinforced sequence $$\hat{X}_1, \hat{X}_2 , \ldots $$ as follows. If $$\varepsilon _n=0$$ , then $$ \hat{X}_n$$ is a uniform random sample from $$\hat{X}_1, \ldots , \hat{X}_{n-1}$$ ; otherwise $$ \hat{X}_n$$ is a new independent copy of X. The purpose of this work is to compare the scaling exponent of the usual random walk $$S(n)=X_1+\cdots + X_n$$ with that of its step reinforced version $$\hat{S}(n)=\hat{X}_1+\cdots + \hat{X}_n$$ . Depending on the tail of X and on asymptotic behavior of the sequence $$(\varepsilon _n)$$ , we show that step reinforcement may speed up the walk, or at the contrary slow it down, or also does not affect the scaling exponent at all. Our motivation partly stems from the study of random walks with memory, notably the so-called elephant random walk and its variations.

中文翻译：

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阶跃强化随机游走的缩放指数

令 $$X_1, X_2, \ldots $$ 是某个真实随机变量 X 的 iid 副本。对于任何确定性的 $$\varepsilon _2, \varepsilon _3, \ldots $$ in $$\{0,1\}$$ ，由 HA Simon 引入的基本算法产生了一个增强序列 $$\hat{X}_1, \hat{X}_2 , \ldots $$ 如下。如果 $$\varepsilon _n=0$$ ，则 $$ \hat{X}_n$$ 是来自 $$\hat{X}_1, \ldots , \hat{X}_{n-1 的统一随机样本}$$; 否则 $$ \hat{X}_n$$ 是 X 的一个新的独立副本。这项工作的目的是将通常的随机游走 $$S(n)=X_1+\cdots + X_n$$ 的缩放指数与它的步骤增强版本 $$\hat{S}(n)=\hat{X}_1+\cdots + \hat{X}_n$$ 。根据 X 的尾部和序列 $$(\varepsilon _n)$$ 的渐近行为，我们表明阶梯强化可能会加速步行，或者相反减慢步行速度，或者根本不影响缩放指数。我们的动机部分源于对记忆随机游走的研究，特别是所谓的大象随机游走及其变体。