A Remark on Wick Ordering of Random Variables

TL;DR

This note clarifies that Wick ordering of polynomials of random variables remains invariant under linear transformations of the variables, emphasizing the notation's consistency.

Contribution

It demonstrates that Wick ordered polynomials are unchanged when expressed in terms of linearly related random variables, clarifying the notation's invariance.

Findings

Wick ordering is invariant under linear transformations.

Expressing polynomials in different linear variables does not alter Wick ordering.

The note clarifies the notation used for Wick ordered polynomials.

Abstract

This paper is a small note on the notation $\,:\! q(X)\!:\,$, for the Wick ordering of polynomials $q$ of random variables $X = (X_1,\dotsc,X_n)$, as introduced by Segal in [6]. We argue that expressing $q(X)$ as another polynomial $p$ of a different set of random variables $Y = (Y_1,\dotsc,Y_m)$, does not give rise to a different Wick ordered random variable $\, : \! p(Y) \! : \,$, provided the new random variables $Y_j$ are linear combinations of the $X_i$'s.