Cumulants on the Wiener Space
Ivan Nourdin (PMA), Giovanni Peccati (MODAL'X)
TL;DR
This paper introduces a new method for calculating cumulants of Gaussian field functionals using integration by parts and recursive moments, simplifying traditional diagram-based approaches.
Contribution
It provides explicit formulas for cumulants on Wiener space, replacing complex graph and diagram methods with a more direct analytical approach.
Findings
Derived explicit cumulant formulas for Gaussian functionals
Simplified cumulant computation on Wiener chaos
Replaced diagram-based methods with analytical formulas
Abstract
We combine infinite-dimensional integration by parts procedures with a recursive relation on moments (reminiscent of a formula by Barbour (1986)), and deduce explicit expressions for cumulants of functionals of a general Gaussian field. These findings yield a compact formula for cumulants on a fixed Wiener chaos, virtually replacing the usual "graph/diagram computations" adopted in most of the probabilistic literature.
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