# Scaling exponents of step-reinforced random walks

Abstract : Let $X_1, \ldots$ be i.i.d. copies of some real random variable $X$. For any $\varepsilon_2, \varepsilon_3, \ldots$ 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 $\hat X_1, \cdots, \hat X_{n-1}$; 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 $\varepsilon_j$, 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.
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Preprints, Working Papers, ...
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https://hal.archives-ouvertes.fr/hal-02480479
Contributor : Jean Bertoin <>
Submitted on : Monday, February 17, 2020 - 2:04:28 PM
Last modification on : Saturday, February 22, 2020 - 1:04:04 AM
Long-term archiving on: : Monday, May 18, 2020 - 12:55:41 PM

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• HAL Id : hal-02480479, version 1

### Citation

Jean Bertoin. Scaling exponents of step-reinforced random walks. 2020. ⟨hal-02480479⟩

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