Séminaire de Probabilités

Weighted Sums in Free Probability Theory

by Leonie Neufeld

Salle Pellos (1R2-207) (IMT)

Salle Pellos (1R2-207)



Given free identically distributed self-adjoint random variables $X_1, \dots, X_n$ satisfying certain moment constraints, the free analog of the Berry-Esseen theorem asserts that the distribution of the normalized sum $S_n = n^{-1/2} \sum_{i=1}^{n} X_i$ converges weakly to Wigner's semicircle law with a rate of convergence of order $n^{-1/2}$ measured with respect to the Kolmogorov distance. Replacing the sum $S_n$ by the weighted sum $S_\theta = \sum_{i=1}^n \theta_i X_i$ for certain vectors $\theta=(\theta_1, \dots, \theta_n)$ taken from the unit sphere, we show that the rate of convergence to Wigner's semicircle law can be improved to the order $n^{-1}$. This establishes a free analog of a result proven by Klartag and Sodin in 2012 in classical probability theory.