On a probabilistic generalization of Taylor's theorem
For a distribution function \(F\) with support \((0, \infty)\) and \(0 < {\mathcal E}_ F X < \infty\), the equilibrium distribution \(TF\) is defined as \(TF(x) = {1 \over {\mathcal E}_ F X} \int^ \infty_ 0 F(u) du\); write \(F_ 1(x) = TF(x)\), \(F_ n(x) = TF_{n - 1} (x)\), \(n \geq 2\), when these exist. Suppose \(0 < {\mathcal E}_ F X^ n < \infty\) and \(g\) is a real-valued function having \((n - 1)\)th derivative on the interval \(I_ F\); assume that \((d/dx)^{n - 1} g(x)\) is absolutely continuous on \(I_ F\) and is the indefinite integral of a Borel measurable function \(h\) such that \({\mathcal E}_{F_ n} | h (X) | < \infty\). Then \({\mathcal E}_ F | g(x) | < \infty\) and the following probabilistic generalization of Taylor's theorem holds: \[ {\mathcal E}_ F g(X) =\sum^{n - 1}_{k = 0} \left. \frac{{\mathcal E}_ F X^ k}{k!} (d/dx)^ kg(x) \right|_{x = 0} + \frac{{\mathcal E}_ F X^ n}{n!} {\mathcal E}_{F_ n} h(X). \] {}.
probabilistic generalization of Taylor's theorem
scientific article; zbMATH DE number 571687