Probabilistic Taylor-Type Expansions of Functions

Taylor–Young and Maclaurin series are widely used for approximating smooth functions around a given point. This study investigates a unified stochastic framework for Taylor-type expansions of functions by means of independent random variables. The proposed probabilistic expansions of a function are able to incorporate evaluations of derivatives at different points, leading to a global approach. Exact expansions are obtained for any order of available derivatives, and such Taylor-type expansions enable the statistical inference of the remainder terms. It appears that the traditional Taylor–Young and Maclaurin series are particular cases of the proposed approach thanks to the Dirac probability measure, and guidelines for using Taylor series have been enhanced. Different ways of choosing the optimal distributions of random variables are provided, particularly when truncations are applied. Numerical comparisons are provided as well.