  In this paper we consider a sum of modified Bessel functions of the first
kind of which particular case is used in the study of Kanter's sharp modified
Bessel function bound for concentrations of some sums of independent symmetric
random vectors. We present some monotonicity and convexity properties for that
sum of modified Bessel functions of the first kind, as well as some Tur\'an
type inequalities, lower and upper bounds. Moreover, we point out an error in
Kanter's paper [Ka] and at the end of the paper we pose an open problem, which
may be of interest for further research.
