Quantitative Local Central Limit Theorems

2017-01-07

The central limit theorem is at the heart of probability theory. It shows states that the fluctuations of a large class of models become normally distributed. In this research area I am interested in quantifying the central limit theorem i.e. determining the rate of convergence. I am in particular interested in local versions of the central limit theorem i.e. theorems that show that not only the law of the fluctuations becomes normally distributed, but also probabilities of elementary events or densities converge too.

I am interested in local central limit theorems because of two reasons. The first reason is that there is a nice link between quantitative local central limit theorems and asymptotic enumerate combinatorics. Simplified, if one has a local central limit theorem for a particular sum of independent random variables \(\sum_{i=1}^N X_i\), then one gets an asymptotic formula for a particular object from enumerative combinatorics. If one can additionally quantify the local central limit theorem, one gets estimates of the error term in the asymptotic formula for the object of enumerative combinatorics. Using this approach has an advantage and a disadvantage. The advantage is that due to the robust nature of the central limit theorem, on is able to derive asymptotic formulas for a lot of different objects from asymptotic enumerative combinatorics with this approach. The disadvantage is that this approach almost never will yield the best possible bounds on the error terms in the asymptotic.

Fur further reading explains the link between quantitative local central limit theorems and asymptotic enumerative combinatorics I want to refer to the article with Stephen de Salvo:

A robust quantitative local central limit theorem with applications to enumerative combinatorics and random combinatorial structures.

The second reason for my interest in local central limit theorems is that they can be used to show strict convexity of the free energy in certain spin-systems. The strict convexity of the free energy seems yields several nice properties for the underlying spins systems. For example, it shows the absence of phase transitions and is the single-most important property in order to derive a quantitative hydrodynamic limit (see also the research area hydrodynamic limit). For further reading I want to refer to my PhD thesis:

Equilibrium dynamics of continuous unbounded spin systems.

and the Article by Grunewald, Otto, Villani, Westdickenberg

A two-scale approach to logarithmic Sobolev inequalities and the hydrodynamic limit. Having this connection in mind, it is not surprising that deducing a a quantitative local central limit (up to the second derivative) was one of the key elements in my article with Felix Otto

the article with Max Fathi

and the article