Self-normalized Sums in Free Probability Theory

TL;DR

This paper establishes the convergence of self-normalized sums of free self-adjoint variables to the semicircle law, providing rates of convergence and extending classical self-normalized limit theorems into free probability.

Contribution

It introduces free probability analogs of self-normalized limit theorems and estimates convergence rates in different boundedness settings.

Findings

Convergence to Wigner's semicircle law for free self-normalized sums.

Rate of convergence is approximately n^{-1/2} with a logarithmic factor for bounded variables.

Rate of convergence is approximately n^{-1/4} for unbounded variables.

Abstract

We show that the distribution of self-normalized sums of free self-adjoint random variables converges weakly to Wigner's semicircle law under appropriate conditions and estimate the rate of convergence in terms of the Kolmogorov distance. In the case of free identically distributed self-adjoint bounded random variables, we retrieve the standard rate of order $n^{-1/2}$ up to a logarithmic factor, whereas we obtain a rate of order $n^{-1/4}$ in the corresponding unbounded setting. These results provide free versions of certain self-normalized limit theorems in classical probability theory.