  Renyi's "thinning" operation on a discrete random variable is a natural
discrete analog of the scaling operation for continuous random variables. The
properties of thinning are investigated in an information-theoretic context,
especially in connection with information-theoretic inequalities related to
Poisson approximation results. The classical Binomial-to-Poisson convergence
(sometimes 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 provided for this limit, and nonasymptotic bounds are
also established. This development parallels, in part, the development of
Gaussian inequalities leading to the information-theoretic version of the
central limit theorem. In particular, a "thinning Markov chain" is introduced,
and it is shown to play a role analogous to that of the Ornstein-Uhlenbeck
process in connection to the entropy power inequality.
