Self-normalized Sums in Free Probability Theory

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 $n^{-1\mathrm{/}2}$ up to a logarithmic factor, whereas we obtain a rate of order $n^{-1\mathrm{/}4}$ in the corresponding unbounded setting. These results provide free versions of certain self-normalized limit theorems in classical probability theory.