A short proof of Isserlis' theorem

TL;DR

We show that Isserlis' theorem follows from the invariant tensor theorem for isotropic tensors, linking Gaussian moments to isotropy via a constant $c_k({\bf X})$. For isotropic ${\bf X}$ one has $E(\prod_{i=1}^{2k}{\bf a}_i^T{\bf X})= c_k({\bf X})\sum_{p\in PP(2k)}\prod_{(l,r)\in p} E({\bf a}_l^T{\bf X}\, {\bf a}_r^T{\bf X})$, and when ${\bf X}$ is Gaussian with zero mean, $c_k({\bf X})=1$, yielding the classical Isserlis formula $E(Y_1\cdots Y_{2k})=\sum_{p\in PP(2k)}\prod_{(l,r)\in p} E(Y_l Y_r)$. The paper also discusses connections to exotic aromatic Butcher series and introduces lianas as a generalization, suggesting broader algebraic and numerical implications for equivariant tensorial moment decompositions.

Abstract

We show that Isserlis' theorem follows as a corollary to the invariant tensor theorem for isotropic tensors.

Theorems & Definitions (8)

• Theorem 1: Isserlis
• Theorem 2: Invariant tensor theorem for isotropic tensors
• proof
• Lemma 1
• proof
• Theorem 3
• proof
• Theorem 4