Cumulants in noncommutative probability I. Noncommutative Exchangeability Systems

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 cumulants and various q-cumulants.

30 pages, AMS-LaTeX; a few minor corrections

References in corpus (2)

• Cumulants in noncommutative probability II. Generalized Gaussian random variables
• Cumulants in Noncommutative Probability Theory III. Creation- and annihilation operators on Fock spaces

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