The Lieb conjecture on the monotonicity of Shannon’s entropy is true

You need to know: Basic probability theory: random variable, its support, probability density function, expectation (denoted $E[.]$ ), independent random variables, identically distributed random variables.

Background: For a random variable X with probability density function f and support $S\subset{\mathbb R}$ , its Shannon entropy is $\text{Ent}(X) = -\int_S f(x) \log f(x) dx$ . Random variable X is square-integrable if $E[X^2]<\infty$ .

The Theorem: On 4th September 2003, Shiri Artstein, Keith Ball, Franck Barthe, and Assaf Naor submitted to the Journal of the AMS a paper in which they proved that, for any sequence $X_1, X_2, \dots$ of independent and identically distributed square-integrable random variables, the entropy of the normalised sum $\text{Ent} \left(\frac{X_1 + \dots + X_n}{\sqrt{n}}\right)$ is a non-decreasing function of n.

Short context: For sequence $X_1, X_2, \dots$ as above, it is known that normalised sums $Y_n = \frac{1}{\sqrt{n}}\sum\limits_{i=1}^n X_i$ converge to normal distribution. In 1949 Shannon proved that $\text{Ent}(Y_2) \geq \text{Ent}(Y_1)$ . In 1978, Lieb conjectured that in fact $\text{Ent}(Y_{n+1}) \geq \text{Ent}(Y_n)$ for all n, which could be interpreted that $Y_n$ becomes closer and closer to the normal distribution (the one with maximal entropy) at every step. This conjecture was open even for $n=2$ . The Theorems proves it in full.