Information-singular random processes

Abstract

Random processes that generate information slower than linearly with time are termed information-singular. The study of information-singularity contributes to a more thorough understanding of the mathematical nature of information generation. Specifically, it elucidates the manner in which generation of information by a time series is critically dependent on the detailed behavior of the sample functions of its spectral representation. The main theorem states that any random sequence whose spectral representation has stationary independent increments with no Brownian motion component is information-singular in the mean-squared sense. The concept of information-singularity can be construed as a means for discriminating between deterministic and nondeterministic processes. It is felt that information-singularity fulfills this discriminating function in a physically more satisfying manner than does the classical Hilbert space theory of linear and nonlinear prediction. The desire for a still more satisfying discriminant motivates investigation of the class of random processes that retain their information-singularity even when corrupted by additive noise. In the case of strictly stationary processes, the discussion focuses on the relationship between information-singularity and zero entropy. Lastly, some alternative definitions of information-singularity are considered and several open problems are identified.