Mathematische Zeitschrift

, Volume 248, Issue 1, pp 67–100

Cumulants in noncommutative probability theory I. Noncommutative exchangeability systems


DOI: 10.1007/s00209-004-0653-0

Cite this article as:
Lehner, F. Math. Z. (2004) 248: 67. doi:10.1007/s00209-004-0653-0


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.

Copyright information

© Springer-Verlag Berlin Heidelberg 2004

Authors and Affiliations

  1. 1.Institut für Mathematik CTechnische Universität GrazGrazAustria