Ulam's History-dependent Random Adding Process

TL;DR

This paper studies the asymptotic behavior of Ulam's history-dependent random sequence, revealing convergence properties, moments, and auto-covariance, and introduces related processes with stochastic change times.

Contribution

It provides new insights into the asymptotic properties and moments of Ulam's process, and extends the model to processes with random change times.

Findings

n^{-2} rac{1}{n} o a non-degenerate limit

Explicit formulas for second moments involving hyperbolic sine

Processes with random change times exhibit similar properties

Abstract

Ulam has defined a history-dependent random sequence of integers by the recursion $X_{n+1}$ $= X_{U(n)}+X_{V(n)}, n \geqslant r$ where $U(n)$ and $V(n)$ are independently and uniformly distributed on $\{1,\dots,n\}$, and the initial sequence, $X_1=x_1,\dots,X_r=x_r$, is fixed. We consider the asymptotic properties of this sequence as $n \to \infty$, showing, for example, that $n^{-2} \sum_{k=1}^n X_k$ converges to a non-degenerate random variable. We also consider the moments and auto-covariance of the process, showing, for example, that when the initial condition is $x_1 =1$ with $r =1$, then $\lim_{n\to \infty} n^{-2} E X^2_n = (2 \pi)^{-1} \sinh(\pi)$; and that for large $m < n$, we have $(m n)^{-1} E X_m X_n \doteq (3 \pi)^{-1} \sinh(\pi).$ We further consider new random adding processes where changes occur independently at discrete times with probability $p$, or where changes occur continuously at jump times of an independent Poisson process. The processes are shown to have properties similar to those of the discrete time process with $p=1$, and to be readily generalised to a wider range of related sequences.