On a probability space let real \(A,D,X\in L^4\), assume \(X\) is independent of \((A,D)\), and suppose \(\mathbb {E}X=\mathbb {E}X^3=0\). Then \(\mathbb {E}(A+XD)^2=\mathbb {E}A^2+\mathbb {E}X^2\mathbb {E}D^2\) and \(\mathbb {E}(A+XD)^4=\mathbb {E}A^4+6\mathbb {E}X^2\mathbb {E}[A^2D^2]+\mathbb {E}X^4\mathbb {E}D^4\).

Let \(A,D\in L^4(\mu )\) be real functions on an arbitrary measure space. Then \((\int A^2D^2\, d\mu )^2\le (\int A^4\, d\mu )(\int D^4\, d\mu )\).

Let \((X_j)_{j\in [n]}\) be mutually independent, almost everywhere measurable real functions on a measure space \((\Omega ,\mu )\), let \(s\subseteq [n]\) and \(i\notin s\), and let \(F,G\) be real coefficient families on subsets of \([n]\). Then \(X_i\) is independent of the pair \((\sum _{T\subseteq s}F(T)\prod _{j\in T}X_j,\sum _{T\subseteq s}G(T)\prod _{j\in T}X_j)\).

Let \(X_1,\ldots ,X_n\in L^4\) be mutually independent real random variables on a probability space, with \(\mathbb {E}X_i=\mathbb {E}X_i^3=0\) and \(\mathbb {E}X_i^4\le B(\mathbb {E}X_i^2)^2\) for every \(i\), where \(B\ge 1\). If \(F(X)=\sum _{S\subseteq [n]}F(S)\prod _{i\in S}X_i\) has coefficients zero whenever \(|S|\gt k\), then \(F(X)\in L^4\) and \(\mathbb {E}F(X)^4\le \max (B,9)^k(\mathbb {E}F(X)^2)^2\).