On a random model of forgetting

Abstract

Georgiou, Katkov and Tsodyks considered the following random process. Let ${\mathit{x}}_1,{\mathit{x}}_2,\dots$ be an infinite sequence of independent, identically distributed, uniform random points in $[0,1]$. Starting with $\mathit{S}=\{0\}$, the elements ${\mathit{x}}_{\mathit{k}}$ 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 $\mathit{S}(1,\mathit{n})$ denote the content of S after the first n elements ${\mathit{x}}_{\mathit{k}}$ join. Simulations suggest that the size $|\mathit{S}(1,\mathit{n})|$ of S at time n is typically close to $\mathit{n}/\mathit{e}$. Here we first give a rigorous proof that this is indeed the case, and that in fact the symmetric difference of $\mathit{S}(1,\mathit{n})$ and the set $\{{\mathit{x}}_{\mathit{k}}\ge 1-1/\mathit{e}:1\le \mathit{k}\le \mathit{n}\}$ is of size at most $\tilde{\mathit{O}}(\sqrt{\mathit{n}})$ with high probability. Our main result is a more accurate description of the process implying, in particular, that as n tends to infinity ${\mathit{n}}^{-1/2}(|\mathit{S}(1,\mathit{n})|-\mathit{n}/\mathit{e})$ converges to a normal random variable with variance $3{\mathit{e}}^{-2}-{\mathit{e}}^{-1}$. We further show that the dynamics of the symmetric difference of $\mathit{S}(1,\mathit{n})$ and the set $\{{\mathit{x}}_{\mathit{k}}\ge 1-1/\mathit{e}:1\le \mathit{k}\le \mathit{n}\}$ converges with proper scaling to a three-dimensional Bessel process.