Cumulants in noncommutative probability I. Noncommutative Exchangeability Systems

Details Cumulants in noncommutative probability I. Noncommutative Exchangeability Systems Cumulants in noncommutative probability I. Noncommutative Exchangeability Systems

2002-10-29

arxiv.org/abs/math/0210442v3

Math. Z. 248 (2004), 67--100

Mathematics

Combinatorics

30 pages, AMS-LaTeX; a few minor corrections

Scientific paper

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.

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