T-Moments and T-Cumulants

Probability distributions are characterized by their moments and cumulants under some very broad assumptions. We argue that using the tensor products of vectors leads to an intuitive and natural way to deal with higher-order moments and cumulants for multivariate distributions, as will be shown in this chapter. We use characteristic and cumulant generating functions to derive the basic theory of T-moments and T-cumulants. We consider the joint moments and cumulants for a collection of multiple random variables, keeping those having the same order together in a vector. These vector-valued quantities have a tensor-product structure. After listing the basic properties of both moments and cumulants, the multivariate Faa di Bruno’s formula is applied to derive relationships between them, namely cumulants are expressed in terms of moments and vice versa. We also provide results concerning the cumulants of products in terms of products of cumulants, conditional cumulants, and cumulants of the log-likelihood function among others. The importance of Bell polynomials in practical applications of the formulae is pointed out in several cases. In our treatment, both moments and cumulants are strongly connected to the T-differential operator \(D_{\mathbf {x}}^{\otimes }\), so that we follow the notations inherited from calculus as well as the traditional Kendall–Stuart notations.