On multivariable cumulant polynomial sequences with applications

Abstract

A new family of polynomials, called cumulant polynomial sequence, and its extension to the multivariate case is introduced relying on a purely symbolic combinatorial method. The coecients are cumulants, but depending on what is plugged in the indeterminates, moment sequences can be recovered as well. The main tool is a formal generalization of random sums, when a not necessarily integer-valued multivariate random index is considered. Applications are given within parameter estimations, Levy processes and random matrices and, more generally, problems involving multivariate functions. The connection between exponential models and multivariable Sheer polynomial sequences oers a dierent viewpoint in employing the method. Some open problems end the paper.