Quantitative concentration inequalities for empirical measures on non-compact spaces

In the second seminar of Hidden Gems in probability, we will discuss how the techniques in showing the Sanov's theorems in Large Deviation combined with results about measure concentration and transportation distances to derive concentration estimates for empirical measure on non-compact spaces. As an application, we will discuss how those estimates can be applied to provide some error bounds for particle simulations in mean-field problem by first deriving regularity and moments estimates for solutions of McKean-Vlasov equation, and then using the coupling argument to show the convergence of equilibrium in large time. The title and part of the material are taken from a recent paper by Bolley, Guillin, and Villani.