Tree based functional expansions for Feynman--Kac particle models (Q1024905)

This paper goes deeper into the theory and applications developed in the magnificent book by the first author [Feynman-Kac formulae: Genealogical and interacting particle systems with applications. Probability and its applications. New York, NY: Springer. (2004; Zbl 1130.60003)]. It aims to provide exact, nonasymptotic, tree-based functional representations of particle block distributions. These distributions as well as the corresponding normalized empirical occupation measures can be expanded polynomially with respect to \(N^{-1}\), where \(N\) stands for the total population size. The proofs rely on an original combinatorial analysis on a special class of trees and forests that parametrize the trajectories of interacting particle systems. The results include an extension of the Wick product formula to these systems. They also provide refined nonasymptotic propagation of chaos-type properties, as well as sharp \(L^p\)-mean error bounds, and laws of large numbers for \(U\)-statistics.