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Cumulants in noncommutative probability I. Noncommutative Exchangeability Systems

Franz Lehner

Published 2002-10-29, updated 2004-09-02Version 3

Cumulants linearize convolution of measures. We use a formula of Good to define noncommutative cumulants in a very general setting.It turns out that the essential property needed is exchangeability of random variables. Roughly speaking the formula says that cumulants are moments of a certain ``discrete Fourier transform'' of a random variable. This provides a simple unified method to understand the known examples of cumulants, like classical, free cumulants and various q-cumulants.

Comments: 30 pages, AMS-LaTeX; a few minor corrections
Journal: Math. Z. 248 (2004), 67--100
Categories: math.CO, math.OA
Subjects: 46L53, 05A18
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