A Remark on Wick Ordering of Random Variables

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.