Thinning and information projections

The law of thin numbers is a Poisson approximation theorem related to the thinning operation. We use information projections to derive lower bounds on the information divergence from a thinned distribution to a Poisson distribution. Conditions for the existence of projections are given. If an information projection exists it must be an element of the associated exponential family. Exponential families are used to derive lower bounds on information divergence and lower bounds on the rate of convergence in the law of thin numbers. A method of translating results related to Poisson distributions into results related to Gaussian distributions is developed and used to prove a new non-trivial result related to the central limit theorem.