Let \((X_i)_{i\in I}\) be mutually independent real random variables on a probability space, with \(I\) finite, \(X_i\in L^2\), \(\mathbb {E}X_i=0\), and \(\mathbb {E}X_i^2=1\) for every \(i\). For \(S,T\subseteq I\), \(\mathbb {E}[(\prod _{i\in S}X_i)(\prod _{i\in T}X_i)]=1\) if \(S=T\), and \(0\) otherwise.

Let \(X_1,\ldots ,X_n\) be mutually independent real random variables on a probability space, each in \(L^4\). Every monomial \(\prod _{i\in S}X_i\), \(S\subseteq [n]\), belongs to \(L^4\).

Let \(X_1,\ldots ,X_n\) be mutually independent real random variables on a probability space, each in \(L^4\), centered, and with second moment \(1\). For real coefficient families \(F,G\) indexed by subsets of \([n]\), writing \(F(X)=\sum _S F(S)\prod _{i\in S}X_i\) and similarly for \(G\), one has \(\mathbb {E}[F(X)G(X)]=\sum _S F(S)G(S)\).

Let \(X_1,\ldots ,X_n\) be mutually independent real random variables on a probability space, each in \(L^4\), centered, and with second moment \(1\). For \(F(X)=\sum _{S\subseteq [n]}F(S)\prod _{i\in S}X_i\) with real coefficients, \(\mathbb {E}[F(X)^2]=\sum _S F(S)^2\).

Let \(X_1,\ldots ,X_n\) be mutually independent real random variables on a probability space, each in \(L^4\), centered, and with second moment \(1\). For arbitrary real coefficients \(F(S)\), \(\mathbb {E}[\sum _{S\subseteq [n]}F(S)\prod _{i\in S}X_i]=F(\varnothing )\).

Let \(F\) be a real coefficient family indexed by subsets of \([n]\), let \(X_i:\Omega \to \mathbb {R}\), and let \(s\subseteq [n]\). If \(F(T)=0\) whenever \(T\nsubseteq s\), then for every \(\omega \), \(\sum _{T\subseteq [n]}F(T)\prod _{j\in T}X_j(\omega )=\sum _{T\subseteq s}F(T)\prod _{j\in T}X_j(\omega )\).

Let \(F\) be real multilinear coefficients on \(n\) inputs and let \(i\in [n]\) satisfy \(F(S)=0\) for every \(S\) containing \(i\). If functions \(X_j,Y_j:\Omega \to \mathbb {R}\) agree for every \(j\ne i\), then \(\sum _S F(S)\prod _{j\in S}X_j=\sum _S F(S)\prod _{j\in S}Y_j\) as functions on \(\Omega \).