Noga Alon, Dor Elboim, Allan Sly

Research output: Contribution to journalArticlepeer-review


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.

Original languageEnglish (US)
Pages (from-to)2190-2207
Number of pages18
JournalAnnals of Applied Probability
Issue number2
StatePublished - Apr 2024

All Science Journal Classification (ASJC) codes

  • Statistics and Probability
  • Statistics, Probability and Uncertainty


  • Memory process


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