The Ergodic Theory of Discrete Sample Paths

This book is about finite-alphabet stationary processes, which are important in physics, engineering, and data compression. The focus is on the combinatorial properties of typical finite sample paths drawn from a stationary, ergodic process. A primary goal, only partially realized, is to develop a theory based directly on sample path arguments with minimal appeals to the probability formalism. A secondary goal is to give a careful presentation of the many models for stationary finite-alphabet processes that have been developed in probability theory, ergodic theory, and information theory.It has emphasis on recent combinatorial results about sample paths. It presents a careful treatment of many models found to be useful in engineering. It includes applications of entropy ideas to coding, sample path structure, distribution estimation, recurrence times, waiting times, and prefix trees. It covers simplification, adaptation, and updating to the process setting of Ornstein isomorphism theory.