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Bounds for an integral of the modified Bessel function of the first kind and expressions involving it (2007.11300v1)
Published 22 Jul 2020 in math.CA
Abstract: Simple upper and lower bounds are obtained for the integral $\int_0x\mathrm{e}{-\gamma t}t\nu I_\nu(t)\,\mathrm{d}t$, $x>0$, $\nu>-\frac{1}{2}$, $0<\gamma<1$. 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.