Cumulants in noncommutative probability I. Noncommutative   Exchangeability Systems
  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. 
Franz Lehner