Thinning and the Law of Small Numbers

The "thinning" operation on a discrete random variable is the natural discrete analog of scaling a continuous variable, i.e., multiplying it by a constant. We examine the role and properties of thinning in the context of information-theoretic inequalities for Poisson approximation. The classical Binomial-to-Poisson convergence, often referred to as the "law of small numbers," is seen to be a special case of a thinning limit theorem for convolutions of discrete distributions. A rate of convergence is also provided for this limit. A Nash equilibrium is established for a channel game, where Poisson noise and a Poisson input are optimal strategies. Our development partly parallels the development of Gaussian inequalities leading to the information- theoretic version of the central limit theorem.