Entropy and the law of small numbers

In this paper, we give an elementary information theoretic proof of some Poisson approximation inequalities for sums of discrete random variables. These can be thought of as "maximum entropy" statements in that, under suitable conditions, the distribution of the sum converges to the distribution which has "maximal entropy" within an appropriate class. We also outline a general method for obtaining corresponding bounds when approximating the distribution of a sum of general discrete random variables by an infinitely divisible distribution.