On the multiplication of free N-tuples of noncommutative random variables
Let $a_{1},\ldots ,a_{n},b_{1},\ldots ,b_{n}$ be random variables in a noncommutative probability space, such that $\{a_{1},\ldots ,a_{n}\}$ is free from $\{b_{1},\ldots ,b_{n}\}$. We show how the joint distribution of the n-tuple $(a_{1}b_{1},\ldots ,a_{n}b_{n})$ can be described in terms of the jo...
Gespeichert in:
| Hauptverfasser: | , |
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| Dokumenttyp: | Article (Journal) |
| Sprache: | Englisch |
| Veröffentlicht: |
1996
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| In: |
American journal of mathematics
Year: 1996, Jahrgang: 118, Heft: 4, Pages: 799-837 |
| ISSN: | 1080-6377 |
| Online-Zugang: | Verlag, lizenzpflichtig, Volltext: https://www.jstor.org/stable/25098492
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| Verfasserangaben: | Alexandru Nica, Roland Speicher |
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| 520 | |a Let $a_{1},\ldots ,a_{n},b_{1},\ldots ,b_{n}$ be random variables in a noncommutative probability space, such that $\{a_{1},\ldots ,a_{n}\}$ is free from $\{b_{1},\ldots ,b_{n}\}$. We show how the joint distribution of the n-tuple $(a_{1}b_{1},\ldots ,a_{n}b_{n})$ can be described in terms of the joint distributions of $(a_{1},\ldots ,a_{n})$ and $(b_{1},\ldots ,b_{n})$, 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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