## Abstract

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 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 {x_{k} ≥ 1 - 1/e : 1 ≤ k ≤ n} converges with proper scaling to a three-dimensional Bessel process.

Original language | English (US) |
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Pages (from-to) | 2190-2207 |

Number of pages | 18 |

Journal | Annals of Applied Probability |

Volume | 34 |

Issue number | 2 |

DOIs | |

State | Published - Apr 2024 |

## All Science Journal Classification (ASJC) codes

- Statistics and Probability
- Statistics, Probability and Uncertainty

## Keywords

- Memory process