Asymptotic properties of chains of infinite order

Abstract

A chain of infinite order consists in a initial condition and a set of transition probabilities which define the probability of appearance of a symbol at each time given the past. Usually, uniqueness and phase transition of chains of infinite order are studied considering uniqueness/non uniqueness of stationary chains, called g-measures. Despite their success, the theory of g-measures naturally excludes some important questions about asymptotic stability of these chains. In this project, we intend to use a different approach to study the stability of chain of infinite order. This approach is based on the notion of dynamic uniqueness recently introduced by Gallesco, Gallo, Takahashi. First, we intend to give a constructive proof for the dynamic phase transition in the Bramson and Kalikow model. Then, we will use the notion of dynamic uniqueness to study the mixing properties of chain of infinite order with positive kernels. (AU)