On a random model of forgetting

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 O˜(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.