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

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dc.creatorNica, Alexandru
dc.creatorSpeicher, Roland
dc.date1996-04-30
dc.date.accessioned2026-07-07T09:13:37Z
dc.date.available2026-07-07T09:13:37Z
dc.descriptionLet $a_{1},...,a_{n}, b_{1},...,b_{n}$ be random variables in some (non-commutative) 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 non-commutative random variables. A different approach to two of these applications is presented by Dan Voiculescu in an Appendix to the paper.
dc.descriptionLaTeX2e, Appendix upon request, to appear in Amer. J. Math
dc.identifierhttps://arxiv.org/abs/funct-an/9604011
dc.identifierhttp://arxiv.org/abs/funct-an/9604011
dc.identifier.urihttp://salesiana.dossiersoluciones.com/handle/123456789/152390
dc.subjectFunctional Analysis
dc.subjectOperator Algebras
dc.subjectQuantum Algebra
dc.titleOn the multiplication of free $n$-tuples of non-commutative random variables
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