Fisher Information, Compound Poisson Approximation, and the Poisson Channel

Fisher information plays a fundamental role in the analysis of Gaussian noise channels and in the study of Gaussian approximations in probability and statistics. For discrete random variables, the scaled Fisher information plays an analogous role in the context of Poisson approximation. Our first results show that it also admits a minimum mean squared error characterization with respect to the Poisson channel, and that it satisfies a monotonicity property that parallels the monotonicity recently established for the central limit theorem in terms of Fisher information. We next turn to the more general case of compound Poisson distributions on the nonnegative integers, and we introduce two new "local information quantities" to play the role of Fisher information in this context. We show that they satisfy subadditivity properties similar to those of classical Fisher information, we derive a minimum mean squared error characterization, and we explore their utility for obtaining compound Poisson approximation bounds.