2020/05/21 by Morgane Austern, Austern, Morgane
Mathematics · #Advanced Operator Algebra Research #FOS: Mathematics #Probability (math.PR) #Random Matrices and Applications #Stochastic processes and statistical mechanics
paper · pdf · doi:10.48550/arxiv.2005.10923
openalex publication_date 2020/05/21 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
The free central-limit theorem, a fundamental theorem in free probability, states that empirical averages of freely independent random variables are asymptotically semi-circular. We extend this theorem to general dynamical systems of operators that we define using a free random variable X coupled with a group of *-automorphims describing the evolution of X. We introduce free mixing coefficients that measure how far a dynamical system is from being freely independent. Under conditions on those coefficients, we prove that the free central-limit theorem also holds for these processes and provide Berry-Essen bounds. We generalize this to triangular arrays and U-statistics. Finally we draw connections with classical probability and random matrix theory with a series of examples.