Asymptotic Mean Stationarity of Sources With Finite Evolution Dimension Ulrich Faigle and Alexander Sch¨ onhuth Mathematisches Institut Zentrum f¨ ur Angewandte Informatik Universit¨ at zu K ¨ oln Weyertal 80, 50931 K¨ oln, Germany faigle@zpr.uni-koeln.de aschoen@zpr.uni-koeln.de Abstract. The notion of the evolution of a discrete random source with finite alphabet is introduced and its behavior under the action of an associated linear evolution operator is studied. Viewing these sources as possibly stable dynamical systems it is proved that all random sources with finite evolution dimension are asymptotically mean stationary, which implies that such random sources have ergodic properties and a well-defined entropy rate. It is shown that the class of random sources with finite evolution dimension properly generalizes the well- studied class of finitary stochastic processes, which includes (hidden) Markov sources as special cases. Keywords. Asymptotic mean, dimension, entropy, ergodic, evolution operator, hidden Markov model, linearly dependent process, Markov chain, observable op- erator model, random source, stable, state generating function, stationary 1 Introduction A central problem of data analysis is learning from sequences that appear to be issued by a random source. In order to admit appropriate learning models, however, the ran- dom source should be such that sampling yields reliable information. As pointed out in Choi et al. [3], for example, most models simply go on the assumption that the random source in question is stationary, which typically is not the case–even when the source is Markov. Moreover, also the theoretical literature usually restricts the study of ergodic and entropic properties to stationary random sources (see, e.g., Han and Kobayashi [7]). Birkhoff’s ergodic theorem, on the other hand, provides a key to a certain converse. One can show that the presence of ergodic properties with respect to bounded measure- ments is equivalent to the seemingly weaker property of asymptotic mean stationarity. Moreover, asymptotically mean stationary sources guarantee the entropy ergodic theo- rem of Shannon-McMillan-Breiman to hold, see [13]. Therefore, it is of both theoretical and practical interest to know which random sources are asymptotically mean stationary (AMS). It is the purpose of the present note to exhibit a large class of random sources to be AMS. We show that this class