Inequalities for an integral involving the modified Bessel function of the first kind

TL;DR

This paper derives simple, tight bounds for an integral involving the modified Bessel function, providing asymptotically accurate estimates useful in probability approximations, especially Stein's method for variance-gamma distributions.

Contribution

It introduces new bounds for the integral of the modified Bessel function, including a generalization, with optimal constants and asymptotic properties, enhancing approximation techniques.

Findings

Established an upper bound valid for all positive x and relevant parameters.

Provided bounds that are tight as x approaches 0 or infinity.

Applied bounds to improve Stein's method for variance-gamma approximation.

Abstract

Simple bounds are obtained for the integral $\int_0^x\mathrm{e}^{-\gamma t}t^\nu I_\nu(t)\,\mathrm{d}t$, $x>0$, $\nu>-1/2$, $0\leq\gamma<1$, together with a natural generalisation of this integral. In particular, we obtain an upper bound that holds for all $x>0$, $\nu>-1/2$, $0\leq\gamma<1$, is of the correct asymptotic order as $x\rightarrow0$ and $x\rightarrow\infty$, and possesses a constant factor that is optimal for $\nu\geq0$ and close to optimal for $\nu>-1/2$. We complement this upper bound with several other upper and lower bounds that are tight as $x\rightarrow0$ or as $x\rightarrow\infty$, and apply our results to derive sharper bounds for some expressions that appear in Stein's method for variance-gamma approximation.