Convergence in the Moment

Originaly posed by T. Stieltjes in the analytical study of continuous fractions, the moment problem motivated Stieltjes to introduce his famous "Stieltjes integral". The moment problem can be simply stated as

"Given the moments of all orders of a body, find the distribution of its mass."

The moment problem has deep connections with analysis and probability theory. It captured the interest of P. Chebyshev and throughout his life. Chebyshev saw this problem primarily as a way of obtaining a certain limit theorem in probability. This was accomplished by A. Markov.

In this project you will investigate how the moment problem connects with probability theory and how it is to this day,

a central tool in the study of convergence of concrete models in probability.

The idea is to develop a broad view on the problem of convergence and to enhance it with applications. Here you may devellop applications of the moment problem to examine the convergence of models in statistical mechanics (point processes, random walks, etc) in the canonical case (i.e. when they admit Poisson/Gaussian limits) or for newer classes of probability limits (Non Gaussian limits: Lamperti regime/Determinantal Point processes).

Prerequisites

Probability II.
Interest in analysis and in the history of mathematics are good qualities for this project.

Resources