Cumulants in noncommutative probability theory I. Noncommutative exchangeability systems

摘要

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.