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Cumulants in noncommutative probability theory I. Noncommutative exchangeability systems

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Abstract.

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 and various q-cumulants.

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  1. Institut für Mathematik C, Technische Universität Graz, Steyrergasse 30, A-8010, Graz, Austria

    Franz Lehner

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  1. Franz Lehner
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Correspondence to Franz Lehner.

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Mathematics Subject Classification (1991): Primary 46L53, Secondary 05A18

Supported by the European Network No HPRN-CT-2000-00116 and the Austrian Science Fund (FWF), Project NoR2-MAT

in final form: 9 September 2003

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Lehner, F. Cumulants in noncommutative probability theory I. Noncommutative exchangeability systems. Math. Z. 248, 67–100 (2004). https://doi.org/10.1007/s00209-004-0653-0

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