History-dependent random processes

Abstract

Ulam has defined a history-dependent random sequence by the recursionXn+1=Xn+XU(n), where (U(n);n≥1) is a sequence of independent random variables withU(n) uniformly distributed on {1, …, n} andX1=1. We introduce a new class of continuous-time history-dependent random processes regulated by Poisson processes. The simplest of these, a univariate process regulated by a homogeneous Poisson process, replicates in continuous time the essential properties of Ulam's sequence, and greatly facilitates its analysis. We consider several generalizations and extensions of this, including bivariate and multivariate coupled history-dependent processes, and cases when the dependence on the past is not uniform. The analysis of the discrete-time formulations of these models would be at the very least an extremely formidable project, but we determine the asymptotic growth rates of their means and higher moments with relative ease.