Moments and Cumulants in Infinite Dimensions with Applications to Poisson, Gamma and Dirichlet-Ferguson Random Measures

TL;DR

This paper explores the combinatorial and algebraic structures underlying moments and cumulants of infinite-dimensional measures like Poisson, Gamma, and Dirichlet-Ferguson, revealing new insights into their representations and symmetries.

Contribution

It introduces a novel combinatorial framework for understanding moments and cumulants in infinite dimensions, especially for Gamma and Dirichlet-Ferguson measures, linking them to symmetric groups and special functions.

Findings

Identifies an intrinsic combinatorial structure in chaos representations of Compound Poisson processes.

Provides a combinatorial interpretation of extended Fock spaces associated with these measures.

Connects the complexity of Gamma measures to Dirichlet-Ferguson measures and their algebraic properties.

Abstract

We show that the chaos representation of some Compound Poisson Type processes displays an underlying intrinsic combinatorial structure, partly independent of the chosen process. From the computational viewpoint, we solve the arising combinatorial complexity by means of the moments/cumulants duality for the laws of the corresponding processes, themselves measures on distributional spaces, and provide a combinatorial interpretation of the associated 'extended' Fock spaces. From the theoretical viewpoint, in the case of the Gamma measure, we trace back such complexity to its 'simplicial part', i.e. the Dirichlet-Ferguson measure, hence to the Dirichlet distribution on the finite-dimensional simplex. We thoroughly explore the combinatorial and algebraic properties of the latter distribution, arising in connection with cycle index polynomials of symmetric groups and dynamical symmetry algebras of confluent Lauricella functions.