On a random model of forgetting

Georgiou, Katkov and Tsodyks considered the following random process. Let $x_1,x_2,\ldots$ 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 $\{x_k\ge 1-1/e: 1 \leq k \leq n \}$ is of size at most $\tilde{O}(\sqrt n)$ with high probability. Our main result is a more accurate description of the process implying, in particular, that as $n$ tends to infinity $n^{-1/2}\big( |S(1,n)|-n/e \big)$ 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 $\{x_k\ge 1-1/e: 1 \leq k \leq n \}$ converges with proper scaling to a three dimensional Bessel process.