History-dependent random processes

Abstract

Ulam has defined a history-dependent random sequence by the recursion X n+1 = Xx + u(n) where (U (n) n ≥ 1) is a sequence of independent random variables with U(n) uniformly distributed on {1, ..., n} and X1=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. © 2008 The Royal Society.