Ergodic and Mixing Properties of Infinite Memory Channels
Roy L. Adler · Proceedings of the American Mathematical Society · 1961
Introduction. A. I. Khinchin [5] states that if an ergodic message space is fed into a channel with finite memory then the output message space is ergodic along with the compound message space of the input with the output.However, Khinchin's notion of finite memory, definition (1) below, is apparently insufficient to yield this result.K. Takano [ó] has been able to establish it by strengthening the definition of finite memory to include both (1) and (2) below.The essential requirement for this theorem, however, is really (3) of which (2) is a special case.Condition(3) expresses that the output of a channel be asymptotically independent from the remote past of the input.The method of proof is an application of a functional form of the notion of ergodicity involving Cesàro convergence of a certain sequence of integrals.In addition this technique can be used to discuss some of the mixing and ergodic properties of the output with respect to the input and the channel.2. Notation and Definitions.2Let (X, X) be a measurable space with X a space of points and X a sigma-field of measurable subsets of X. Usually in information theory X is referred to as an alphabet and is a finite set of points.We shall make no such restriction here.Con-