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 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.