Self-similarity in an Exchangeable Site-Dynamics Model

Abstract

A system of sites indexed by ${\mathbb N}$ are each assigned an initial fitness value in [0, 1]. Each unit of time an environment is sampled for the system and a proposed fitness, uniform in [0, 1], is sampled for each site, all independent of each other and the past. The environment is good with probability p and is bad otherwise. The fitness of each site is then updated to the maximum or the minimum of its current fitness and the proposed fitness according to whether the environment is good or bad. We study the empirical fitness distribution, a probability-valued Markov process, and prove that it converges in distribution to a unique stationary distribution. Under the stationary distribution the system exhibits a self-similar (typically) non-smooth site-exchangeable behavior whereas the fitness distribution for each site is smooth. Our analysis is done by identifying a class of iterated function systems for which we prove ergodicity and provide a probabilistic representation of the stationary distribution. This also yields a close connection to the fractals literature.