On the multiplication of free N-tuples of noncommutative random variables - Nica, Alexandru; Speicher, Roland (American journal of mathematics, 1996)
Let $a_{1},… ,a_{n},b_{1},… ,b_{n}$ be random variables in a noncommutative probability space, such that $\{a_{1},… ,a_{n}\}$ is free from $\{b_{1},… ,b_{n}\}$. We show how the joint distribution of the n-tuple $(a_{1}b_{1},… ,a_{n}b_{n})$ can be described in terms of the joint distributions of $(a_{1},… ,a_{n})$ and $(b_{1},… ,b_{n})$, by using the combinatorics of the n-dimensional R-transform. We point out a few applications that can be easily derived from our result, concerning the left-and-right translation with a semicircular element (see Sections 1.6-1.10) and the compression with a projection (see Sections 1.11-1.14) of an n-tuple of noncommutative random variables. A different approach to two of these applications is presented by Dan Voiculescu in an Appendix to the paper.
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