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

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 $x\to \infty$. 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.

18 pages