  An "entropy increasing to the maximum" result analogous to the entropic
central limit theorem (Barron 1986; Artstein et al. 2004) is obtained in the
discrete setting. This involves the thinning operation and a Poisson limit.
Monotonic convergence in relative entropy is established for general discrete
distributions, while monotonic increase of Shannon entropy is proved for the
special class of ultra-log-concave distributions. Overall we extend the
parallel between the information-theoretic central limit theorem and law of
small numbers explored by Kontoyiannis et al. (2005) and Harremo\"es et al.\
(2007, 2008). Ingredients in the proofs include convexity, majorization, and
stochastic orders.
