Let \(\psi \colon \mathbb R\to \mathbb R\) be \(C^4\) and let \(C\ge 0\) satisfy \(|\psi ^{(4)}(t)|\le C\) for all real \(t\). For all real \(a,b\), \(\left|\psi (a+b)-\sum _{j=0}^3\psi ^{(j)}(a)b^j/j!\right|\le Cb^4/24\).

Let \(Z\in L^4(\mu )\) be real-valued on a probability space \((\Omega ,\mu )\), and let \(\psi \colon \mathbb R\to \mathbb R\) be \(C^4\). If some \(C\ge 0\) bounds \(|\psi ^{(4)}(t)|\) for every real \(t\), then \(\psi (Z)\) is integrable under \(\mu \).

Let \(X\) be a real input on a probability space, measurable up to a null set with finite fourth absolute moment, first three moments \(0,1,0\), and fourth moment at most \(9\). For any real function \(\psi \) and real \(a,b\), \(\mathbb E[\sum _{j=0}^3\psi ^{(j)}(a)(bX)^j/j!]=\psi (a)+\tfrac 12\psi ^{(2)}(a)b^2\). Here iterated differentiation assigns zero at a point where the derivative does not exist, so the identity requires no smoothness of \(\psi \).

Let \(X\) be a real input on a probability space, measurable up to a null set with finite fourth absolute moment, first three moments \(0,1,0\), and fourth moment at most \(9\). Let \(\psi \colon \mathbb R\to \mathbb R\) be \(C^4\) and let \(C\ge 0\) satisfy \(|\psi ^{(4)}(t)|\le C\) for all real \(t\). For all real \(a,b\), \(|\mathbb E\psi (a+bX)-\psi (a)-\tfrac 12\psi ^{(2)}(a)b^2|\le (3C/8)b^4\).

Let \(X\) and \(Y\) be real inputs on possibly different probability spaces, each measurable up to a null set with finite fourth absolute moment, first three moments \(0,1,0\), and fourth moment at most \(9\). Let \(\psi \colon \mathbb R\to \mathbb R\) be \(C^4\) and let \(C\ge 0\) satisfy \(|\psi ^{(4)}(t)|\le C\) for all real \(t\). For every real \(a,b\), \(|\mathbb E\psi (a+bX)-\mathbb E\psi (a+bY)|\le (3C/4)b^4\).