On the accuracy of the approximation of the complex exponent by the first terms of its Taylor expansion with applications

TL;DR

This paper develops new bounds for the Taylor approximation of the complex exponential, improving accuracy estimates for characteristic functions and their derivatives, with applications to probability distributions and moment inequalities.

Contribution

It introduces a new bound for the Taylor remainder of the complex exponential and derives sharper moment inequalities, enhancing approximation accuracy for characteristic functions.

Findings

New bound for the Taylor remainder of $e^{ix}$

Sharper upper bounds for the third moment of distributions

Improved uniform bounds for characteristic functions

Abstract

A new bound for the remainder term in the Taylor expansion of the complex exponent $e^{ix}$, $x\in\R$, is proved yielding precise moment-type estimates of the accuracy of the approximation of the characteristic function (the Fourier--Stieltjes transform) of a probability distribution by the first terms of its Taylor expansion. Moreover, a precise upper bound for the third moment of a probability distribution in terms of the absolute third moment is established which sharpens Jensen's inequality. Based on these results, new improved bounds for characteristic functions and their derivatives are obtained that are {\it uniform} in the class of distributions with fixed first three moments.