Revisiting fundamental limit theorems in probability theory

Abstract

We revisit three fundamental limit theorems in probability theory: the strong law of large numbers, the central limit theorem, and Wigner's semicircle law. For the first two theorems, we present elementary proofs including new ones. We also discuss and prove the Hsu--Robbins--Erd\H{o}s theorem. For the semicircle law, we consider random Hermitian matrices with independent upper triangular entries. Under certain conditions including Lindeberg's condition, we characterize convergence to the semicircle distribution in terms of the variances of the entries.