## The Annals of Mathematical Statistics

- Ann. Math. Statist.
- Volume 29, Number 4 (1958), 1209-1220.

### Proof of Shannon's Transmission Theorem for Finite-State Indecomposable Channels

David Blackwell, Leo Breiman, and A. J. Thomasian

#### Abstract

For finite-state indecomposable channels, Shannon's basic theorem, that transmission is possible at any rate less than channel capacity but not at any greater rate, is proved. A necessary and sufficient condition for indecomposability, from which it follows that every channel with finite memory is indecomposable, is given. An important tool is a modification, for some processes which are not quite stationary, of theorems of McMillan and Breiman on probabilities of long sequences in ergodic processes.

#### Article information

**Source**

Ann. Math. Statist., Volume 29, Number 4 (1958), 1209-1220.

**Dates**

First available in Project Euclid: 27 April 2007

**Permanent link to this document**

https://projecteuclid.org/euclid.aoms/1177706452

**Digital Object Identifier**

doi:10.1214/aoms/1177706452

**Mathematical Reviews number (MathSciNet)**

MR118570

**Zentralblatt MATH identifier**

0096.10901

**JSTOR**

links.jstor.org

#### Citation

Blackwell, David; Breiman, Leo; Thomasian, A. J. Proof of Shannon's Transmission Theorem for Finite-State Indecomposable Channels. Ann. Math. Statist. 29 (1958), no. 4, 1209--1220. doi:10.1214/aoms/1177706452. https://projecteuclid.org/euclid.aoms/1177706452