Generalized Wick Decompositions

TL;DR

This paper reviews and generalizes the cumulant and Wick decompositions, extending them to arbitrary functions, providing a unified framework for decomposing products of variables.

Contribution

It introduces a novel generalized decomposition framework that extends traditional cumulant and Wick decompositions to arbitrary functions.

Findings

Unified decomposition framework for arbitrary functions

Extension of cumulant and Wick decompositions

Potential applications in probabilistic and algebraic analysis

Abstract

We review the cumulant decomposition (a way of decomposing the expectation of a product of random variables (e.g. $\mathbb{E}[XYZ]$) into a sum of terms corresponding to partitions of these variables.) and the Wick decomposition (a way of decomposing a product of (not necessarily random) variables into a sum of terms corresponding to subsets of the variables). Then we generalize each one to a new decomposition where the product function is generalized to an arbitrary function.