On the fractional probabilistic Taylor's and mean value theorems

TL;DR

This paper develops fractional probabilistic versions of Taylor's and mean value theorems using fractional calculus, characterizing distributions and extending classical results to non-integer order derivatives.

Contribution

It introduces the nth-order fractional equilibrium distribution and proves fractional Taylor's and mean value theorems, extending classical probabilistic results with fractional calculus tools.

Findings

Characterization of distributions via fractional equilibrium distributions

Development of fractional probabilistic Taylor's theorem

Extension of mean value theorem for nonnegative variables

Abstract

In order to develop certain fractional probabilistic analogues of Taylor's theorem and mean value theorem, we introduce the nth-order fractional equilibrium distribution in terms of the Weyl fractional integral and investigate its main properties. Specifically, we show a characterization result by which the nth-order fractional equilibrium distribution is identical to the starting distribution if and only if it is exponential. The nth-order fractional equilibrium density is then used to prove a fractional probabilistic Taylor's theorem based on derivatives of Riemann-Liouville type. A fractional analogue of the probabilistic mean value theorem is thus developed for pairs of nonnegative random variables ordered according to the survival bounded stochastic order. We also provide some related results, both involving the normalized moments and a fractional extension of the variance, and a formula of interest to actuarial science. In conclusion we discuss the probabilistic Taylor's theorem based on fractional Caputo derivatives.