An introduction to quantitative aspects of the central limit theorem.

In the first part we will review some quantitative aspects of the classical CLT (aka Berry-Esseen theorems). We will also describe some applications (e.g. approximation of pi via Monte Carlo methods).

In the second part we will discuss recent versions of the Berry-Esseen theorem in L^p, from a result of S. Bobkov (2018) for independent random variables to our more recent results for “weakly dependent” random variables like Markov chains, products for random matrices and hyperbolic dynamical systems (all the processes will be described through explicit examples).