Scaling exponents of step-reinforced random walks

• 成果类型: Article
• 署名作者: Bertoin, Jean
• 署名单位: University of Zurich
• 刊物名称: PROBABILITY THEORY AND RELATED FIELDS
• ISSN/ISSBN: 0178-8051
• DOI: 10.1007/s00440-020-01008-2
• 发表日期: 2021
• 页码: 295-315
• 关键词: edge

摘要

Let X-1, X-2,... be i.i.d. copies of some real random variable X. For any deterministic epsilon(2), epsilon(3),... in {0, 1}, a basic algorithm introduced by H.A. Simon yields a reinforced sequence (X) over cap (1), (X) over cap (2),... as follows. If epsilon(n) = 0, then (X) over cap (n) is a uniform random sample from (X) over cap (1), ... , (X) over cap (n-1); otherwise (X) over cap (n) is a newindependent copy of X. The purpose of this work is to compare the scaling exponent of the usual random walk S(n) = X-1 + center dot center dot center dot + X-n with that of its step reinforced version (X) over cap (n) = (X) over cap (1)+ center dot center dot center dot + (X) over cap (n). Depending on the tail of X and on asymptotic behavior of the sequence (epsilon(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.