**Graduate Studies in Mathematics**

Volume: 13;
1996;
249 pp;
Hardcover

MSC: Primary 28; 94;
Secondary 60

Print ISBN: 978-0-8218-0477-3

Product Code: GSM/13

List Price: $50.00

AMS Member Price: $40.00

MAA member Price: $45.00

**Electronic ISBN: 978-1-4704-2071-0
Product Code: GSM/13.E**

List Price: $50.00

AMS Member Price: $40.00

MAA member Price: $45.00

# The Ergodic Theory of Discrete Sample Paths

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*Paul C. Shields*

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.

Features:

- Emphasis on recent combinatorial results about sample paths.
- Careful treatment of many models found to be useful in engineering.
- Applications of entropy ideas to coding, sample path structure, distribution estimation, recurrence times, waiting times, and prefix trees.
- Simplification, adaptation, and updating to the process setting of Ornstein isomorphism theory.

#### Readership

Graduate students and faculty members in mathematics, engineering, statistics, and physics who are interested in measure theory and probability theory.

#### Reviews & Endorsements

A very original book on a topic of current interest written by an experienced author.

-- Zentralblatt MATH

This is a well-written book. Being an expert in the field, the author has chosen his own way of organization and presentation of the material. The result is a unique book with many interesting details for the expert and a text for the graduate student which is well thought out … covers important and useful aspects of probabilistic discrete-sample-paths theory.

-- Mathematical Reviews

This is an interesting and well-written account of the ergodic theory of stationary processes from the viewpoint of entropy theory. This book is a beautiful and very readable treatment of an important part of ergodic theory. It is written by an expert in the field who has made very significant contributions and who has an excellent knowledge of the great relevance of the theory to other fields such as information theory … strongly recommend this book to anyone who is interested in ergodic theory, stochastic processes or information theory.

-- Bulletin of the London Mathematical Society

#### Table of Contents

# Table of Contents

## The Ergodic Theory of Discrete Sample Paths

- Cover Cover11 free
- Title v6 free
- Copyright vi7 free
- Contents vii8 free
- Preface ix10 free
- I: Basic concepts 114 free
- I.1 Stationary processes 114
- I.2 The ergodic theory model 1326
- I.3 The ergodic theorem 3346
- I.4 Frequencies of finite blocks 4356
- I.5 The entropy theorem 5164
- I.6 Entropy as expected value 5669
- I.7 Interpretations of entropy 6679
- I.8 Stationary coding 7992
- I.9 Process topologies 87100
- I.10 Cutting and stacking 103116

- II: Entropy-related properties 121134
- III: Entropy for restricted classes 165178
- IV: B-processes 211224
- Bibliography 239252
- Index 245258
- Back Cover Back Cover1263