Limit Laws for R-diagonal Variables in a Tracial Probability Space

We study the weak convergence of sums of ∗\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${*}$$\end{document}-free, identically distributed tracial R-diagonal variables. The result parallels earlier results about free additive convolution on the real line. In particular, we determine under which conditions an infinitesimal array yields a sequence that converges to a given infinitely divisible tracial R-diagonal distribution. Thus, much of the work concerning sums of free (in the sense of Voiculescu) identically distributed positive random variables can be translated to the tracial R-diagonal context.