Bounds for an integral of the modified Bessel function of the first kind and expressions involving it

TL;DR

This paper derives tight bounds for an integral involving the modified Bessel function of the first kind, which are useful for improving Stein's method in variance-gamma distribution approximations.

Contribution

It provides new simple bounds for a specific integral of the modified Bessel function, enhancing analytical tools for probabilistic approximation methods.

Findings

Bounds are tight as x approaches infinity.

Applied bounds improve variance-gamma approximation techniques.

Facilitates technical advancements in Stein's method.

Abstract

Simple upper and lower bounds are obtained for the integral $\int_0^x\mathrm{e}^{-\gamma t}t^\nu I_\nu(t)\,\mathrm{d}t$, $x>0$, $\nu>-\frac{1}{2}$, $0<\gamma<1$. Most of our bounds for this integral are tight as $x\rightarrow\infty$. We apply one of our inequalities to bound some expressions involving this integral. Two of these expressions appear in Stein's method for variance-gamma approximation, and our bounds will allow for a technical advancement to be made to the method.