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Scaling exponents of step-reinforced random walks


Bertoin, Jean (2020). Scaling exponents of step-reinforced random walks. Probability Theory and Related Fields:online.

Abstract

Let X1,X2,… be i.i.d. copies of some real random variable X. For any deterministic ε2,ε3,… in {0,1}, a basic algorithm introduced by H.A. Simon yields a reinforced sequence X^1,X^2,… as follows. If εn=0, then X^n is a uniform random sample from X^1,…,X^n−1; otherwise 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)=X1+⋯+Xn with that of its step reinforced version S^(n)=X^1+⋯+X^n. Depending on the tail of X and on asymptotic behavior of the sequence (εn), 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.

Abstract

Let X1,X2,… be i.i.d. copies of some real random variable X. For any deterministic ε2,ε3,… in {0,1}, a basic algorithm introduced by H.A. Simon yields a reinforced sequence X^1,X^2,… as follows. If εn=0, then X^n is a uniform random sample from X^1,…,X^n−1; otherwise 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)=X1+⋯+Xn with that of its step reinforced version S^(n)=X^1+⋯+X^n. Depending on the tail of X and on asymptotic behavior of the sequence (εn), 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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Additional indexing

Item Type:Journal Article, refereed, original work
Communities & Collections:07 Faculty of Science > Institute of Mathematics
Dewey Decimal Classification:340 Law
610 Medicine & health
510 Mathematics
Scopus Subject Areas:Physical Sciences > Analysis
Physical Sciences > Statistics and Probability
Social Sciences & Humanities > Statistics, Probability and Uncertainty
Uncontrolled Keywords:Statistics, Probability and Uncertainty, Statistics and Probability, Analysis
Language:English
Date:9 October 2020
Deposited On:15 Dec 2020 15:19
Last Modified:20 Dec 2020 08:53
Publisher:Springer
ISSN:0178-8051
OA Status:Hybrid
Publisher DOI:https://doi.org/10.1007/s00440-020-01008-2

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