Tensor notation and cumulants of polynomials

A modified and extended tensor notation is introduced that is sufficient to cover multivariate moments and cumulants as special cases. Using this notation, two basic identities are given. The first of these expresses generalized cumulants in terms of ordinary cumulants. The second gives the joint cumulant generating function of any polynomial transformation in terms of the cumulants of the original variables. Three applications of the basic identities are given. The first application is concerned with sample cumulants or k-statistics, the second to Edgeworth series and the third to exponential family models.