On the approximation of convex functions with cumulant generating functions (Q857103)

Let \(\mu\) be a measure on \(\mathbb{R}\) and \(M\) be its two-sided Laplace transform. The function \(s\mapsto\ln M(-s)\), \(s\in\mathbb{R}\), is said to be a cumulant generating function (CGF). Theorem. Let \(f: I\to \mathbb{R}\) be continuous convex where \(I\subset\mathbb{R}\) be a bounded interval or \(\mathbb{R}\). There is a measure \(\mu\geq 0\) whose CGF function \(\varphi\) satisfies \(\| f-\varphi\|_{C_0(I)}\leq \ln 2\). The sharp constant of this inequality is \(\geq{\ln 2\over 2}\).