Half independence and half cumulants

The notion of half independence arises in random matrices and quantum groups. This notion is available only for elements of a noncommutative probability space and assumes the existence of all moments. We relate half independence to a certain class of partitions and use it to define an appropriate cumulant generating function and a transform which is closely related to the characteristic function. This leads to a definition of half independent convolution of arbitrary probability measures which is compatible with the distribution of the sum of half independent elements of a noncommutative probability space. We also establish the central limit theorem for half independent convolution of measures with the limit being symmetrized Rayleigh. Cramer's theorem is also established in this set up