Let $\rho(x)$ be a probability density function on $\mathbb{R}$ with prescribed variance $\sigma^2$, so that:
$$\int\_\mathbb{R} \rho(x)\, dx = 1$$
and
$$\int\_\mathbb{R} x^2 \rho(x), dx = \sigma^2$$
Fact: the density function which maximizes the entropy functional
$$S(\rho) = -\int\_\mathbb{R} \rho(x) \log \rho(x)\, dx$$
with the constraints above is the normal distribution 
$$\rho(x) = \frac{1}{\sqrt{2\pi \sigma^2}} e^{-\frac{x^2}{2\sigma^2}}$$
This can be proved using basic techniques from the calculus of variations.


My question: can this be used to prove the central limit theorem? In other words, can one show directly that the limiting distribution of the average of a sequence of i.i.d. random variables maximizes entropy?


Actually, I don't care too much about entropy. I'm mainly interested in the possibility of a variational proof of the central limit theorem.



Central limit theorem via maximal entropy
There's a 1985 article by Derriennic called "Entropie, theoremes limite et marches aleatoires" (entropy, limit theorems and random walks). In it there is a section where the connection between your observation that the Gaussian maximizes entropy (which is attributed to Shannon) and the central limit theorem is discussed.


He begins by discussing a proof (attributed to Pinsker, citing page 20 of his book on Information Stability) that the iterated convolution of a density on a compact group converges to a constant. The proof is based on the fact that the entropy of the sequence of convolutions is monotone.


After this he discusses a work of Csizar (A note on limitimg distributions on topological groups) where the same technique is used to prove the convergence of convolutions of general probabilities on a compact group (not supported on a closed subgroup) to the Haar measure.


Finally, **answering your question**, the proof of the central limit theorem in $\mathbb{R}$ using the idea of entropy monotonicity is attributed to **Linnik**. 


The precise reference being: "**An information-theoretic proof of the central limit theorem with the Lindeberg
condition", Theory of Probability and its applications. 1959, Vol IV, n o 3, 288-299**.


