TY - JOUR

T1 - ON A RANDOM MODEL OF FORGETTING

AU - Alon, Noga

AU - Elboim, Dor

AU - Sly, Allan

N1 - Publisher Copyright:
© Institute of Mathematical Statistics, 2024.

PY - 2024/4

Y1 - 2024/4

N2 - Georgiou, Katkov and Tsodyks considered the following random process. Let x1, x2,... be an infinite sequence of independent, identically distributed, uniform random points in [0, 1]. Starting with S = {0}, the elements xk 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 xk 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 {xk ≥ 1 - 1/e : 1 ≤ k ≤ n} is of size at most Õ(√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 {xk ≥ 1 - 1/e : 1 ≤ k ≤ n} converges with proper scaling to a three-dimensional Bessel process.

AB - Georgiou, Katkov and Tsodyks considered the following random process. Let x1, x2,... be an infinite sequence of independent, identically distributed, uniform random points in [0, 1]. Starting with S = {0}, the elements xk 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 xk 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 {xk ≥ 1 - 1/e : 1 ≤ k ≤ n} is of size at most Õ(√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 {xk ≥ 1 - 1/e : 1 ≤ k ≤ n} converges with proper scaling to a three-dimensional Bessel process.

KW - Memory process

UR - http://www.scopus.com/inward/record.url?scp=85189810730&partnerID=8YFLogxK

U2 - 10.1214/23-AAP2018

DO - 10.1214/23-AAP2018

M3 - ???researchoutput.researchoutputtypes.contributiontojournal.article???

AN - SCOPUS:85189810730

SN - 1050-5164

VL - 34

SP - 2190

EP - 2207

JO - Annals of Applied Probability

JF - Annals of Applied Probability

IS - 2

ER -