Inequalities for integrals of modified Bessel functions and expressions involving them

TL;DR

This paper establishes optimal inequalities for integrals involving modified Bessel functions, providing tight bounds that enhance Stein's method for variance-gamma approximation and advancing related analytical techniques.

Contribution

It introduces new sharp inequalities for Bessel function integrals and applies them to improve bounds in Stein's method for probability approximations.

Findings

Derived best possible constants for inequalities

Obtained uniform bounds for Bessel function integrals

Enabled technical improvements in variance-gamma approximation

Abstract

Simple inequalities are established for some integrals involving the modified Bessel functions of the first and second kind. In most cases, we show that we obtain the best possible constant or that our bounds are tight in certain limits. We apply these inequalities to obtain uniform bounds for several expressions involving integrals of modified Bessel functions. Such expressions occur in Stein's method for variance-gamma approximation, and the results obtained in this paper allow for technical advances in the method. We also present some open problems that arise from this research.