ON A RANDOM MODEL OF FORGETTING
成果类型：
Article
署名作者：
Alon, Noga; Elboim, Dor; Sly, Allan
署名单位：
Princeton University
刊物名称：
ANNALS OF APPLIED PROBABILITY
ISSN/ISSBN：
1050-5164
DOI：
10.1214/23-AAP2018
发表日期：
2024
页码：
2190-2207
关键词：
摘要：
Georgiou, Katkov and Tsodyks considered the following random process. Let x(1), x(2), . . . be an infinite sequence of independent, identically distributed, uniform random points in [0, 1]. Starting with S = {0}, the elements x(k) join S one by one, in order. When an entering element is larger than the current minimum element of S, this minimum leaves S. Let S(1, n) denote the content of S after the first n elements x(k) join. Simulations suggest that the size |S(1, n)| of S at time n is typically close to n/e. Here we first give a rigorous proof that this is indeed the case, and that in fact the symmetric difference of S(1, n) and the set {x(k) >= 1 - 1/e :1 <= k <= n} is of size at most O(root n) with high probability. Our main result is a more accurate description of the process implying, in particular, that as n tends to infinity n(-1/2)(|S (1, n)| - n/e) converges to a normal random variable with variance 3e(-2) - e(-1). We further show that the dynamics of the symmetric difference of S(1, n) and the set {x(k) >= 1 - 1/e : 1 <= k <= n} converges with proper scaling to a three-dimensional Bessel process.
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