On a probabilistic generalization of Taylor's theorem
type: Item
instance of: scholarly article
title: On a probabilistic generalization of Taylor's theorem
zbMATH Open document ID: 0824.60014
DOI: 10.1016/0167-7152(94)90110-4
published in: Statistics \& Probability Letters
publication date: 1995-10-31T00:00:00Z
review text: 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: \[ {\mathcal E}_ F g(X) =\sum^{n - 1}_{k = 0} \left. \frac{{\mathcal E}_ F X^ k}{k!} (d/dx)^ kg(x) \right|_{x = 0} + \frac{{\mathcal E}_ F X^ n}{n!} {\mathcal E}_{F_ n} h(X). \] {}.
Mathematics Subject Classification ID: 60E99
Mathematics Subject Classification ID: 60F99
zbMATH DE Number: 571687
zbMATH Keywords: absolute continuity
zbMATH Keywords: equilibrium distribution
zbMATH Keywords: probabilistic generalization of Taylor's theorem
author: Gwo Dong Lin
reviewed by: Sujit K. Basu
MaRDI profile type: Publication
full work available at URL: 0167-7152(94)90110-4
OpenAlex ID: W2094030021
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description: scientific article; zbMATH DE number 571687