# History-dependent random processes

• Published in 2008
Ulam has defined a history-dependent random sequence by the recursion Xn+1=Xn+XU(n), where (U(n); n[&ge;]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.

## Other information

issn
1364-5021
journal
Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences
number
2093
pages
1105--1124
volume
464

### BibTeX entry

@article{Clifford2008,
abstract = {Ulam has defined a history-dependent random sequence by the recursion Xn+1=Xn+XU(n), where (U(n); n[{\&}ge;]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.},
author = {Clifford, P. and Stirzaker, D.},
issn = {1364-5021},
journal = {Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences},
month = {may},
number = 2093,
pages = {1105--1124},
title = {History-dependent random processes},
url = {http://rspa.royalsocietypublishing.org/content/464/2093/1105},
volume = 464,
year = 2008,
urldate = {2014-06-30}
}