On multivariable cumulant polynomial sequences with applications

TL;DR

This paper introduces multivariable cumulant polynomial sequences using a symbolic combinatorial approach, enabling new insights into moments, cumulants, and applications in stochastic processes and multivariate functions.

Contribution

It presents a novel family of cumulant polynomial sequences and extends them to multivariate cases, linking exponential models with Sheffer polynomial sequences.

Findings

Coefficients are cumulants, recover moments or cumulants depending on substitution.

Applications include parameter estimation, Lévy processes, and random matrices.

Provides a new combinatorial framework for multivariate polynomial sequences.

Abstract

A new family of polynomials, called cumulant polynomial sequence, and its extensions to the multivariate case is introduced relied on a purely symbolic combinatorial method. The coefficients of these polynomials are cumulants, but depending on what is plugged in the indeterminates, either sequences of moments either sequences of cumulants can be recovered. The main tool is a formal generalization of random sums, also with a multivariate random index and not necessarily integer-valued. Applications are given within parameter estimations, L\'evy processes and random matrices and, more generally, problems involving multivariate functions. The connection between exponential models and multivariable Sheffer polynomial sequences offers a different viewpoint in characterizing these models. Some open problems end the paper.