Convergence and asymptotic approximations to universal distributions in probability

Abstract

The limiting distributions of functionals depending on a large number of independent random variables of comparable size are often universal, leading to a vast number of convergence and approximation results. We discuss some general principles that have emerged in recent years. Examples include classical and entropic central limit theorems in classical and free probability, distributions of zeros of random polynomials of high degree and related distributions of algebraic numbers, as well as global and local universality results for spectral distributions of random matrices.