There is a book on the subject: <a href="http://www.worldscientific.com/worldscibooks/10.1142/p341">"Information Theory and The Central Limit Theorem"</a> by Oliver Johnson.
The article by Anshelevich mentioned by Yemon considers the operator $T$ acting on probability densities an corresponding to going from a random variable $X$ to $(X+Y)/\sqrt{2}$ where $Y$ is an independent copy of $X$. The entropy is a Lyapunov function for this transformation which is the simplest example of a renormalization group transformation.
The $N(0,1)$ is a fixed point and it is easy to diagonalize the linearization of $T$ near this fixed point using Wick monomials, i.e., Hermite polynomials. The directions corresponding to 0-th, 1-st and 2-nd moments are expanding (relevant operators) or neutral (marginal operators) while all others are contracting (irrelevant operators). Therefore if one makes the necessary arrangements to fix these moments (e.g. subtracting $N$ times the mean and dividing by $\sqrt{N}$) then one lies on the stable manifold of the Gaussian fiexd point. See the <a href="http://books.google.com/books/about/Theory_of_Probability_and_Random_Process.html?id=tlWOphOFRgwC">textbook on probability  theory</a> by Koralov and Sinai for more details. The generalization of the $T$ map for joint probability distributions of dependent variables, i.e., the renormalization group is explained in the book <a href="http://link.springer.com/book/10.1007%2FBFb0017107">"A Renormalization Group Analysis of the Hierarchical Model in Statistical Mechanics"</a> by Collet and Eckmann. The issue with using this type of nonlinear transformations is that the above diagonalization at a fixed point only gives information about the vicinity of that fixed point. To get results far away, having a Lyapunov function like the entropy is of great importance. This is an active area in physics which investigates generalizations of Zamolodchikov's $c$-"theorem" in conformal field theory. See for instance <a href="http://arxiv.org/abs/1302.0884">this article</a> for a recent review. Entanglement entropy seems to be the Lyapunov function in this setting.