On the multiplication of free n-tuples of non-commutative random variables

Let a1, . . . , an, b1, . . . , bn be random variables in some (non-commutative) probability space, such that {a1, . . . , an} is free from {b1, . . . , bn}. We show how the joint distribution of the n-tuple (a1b1, . . . , anbn) can be described in terms of the joint distributions of (a1, . . . , an) and (b1, . . . , bn), 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 non-commutative random variables. A different approach to two of these applications is presented by Dan Voiculescu in an Appendix to the paper. Research done while this author was on leave at the Fields Institute, Waterloo, and the Queen’s University, Kingston, holding a Fellowship of NSERC, Canada. Supported by a Heisenberg Fellowship of the DFG.