  We give a shorter proof of Kanter's (1976) sharp Bessel function bound for
concentrations of sums of independent symmetric random vectors. We provide
sharp upper bounds for the sum of modified Bessel functions $I_0(x)+I_1(x)$,
which might be of independent interest. Corollaries improve concentration or
smoothness bounds for sums of independent random variables due to Cekanavicius
& Roos (2006), Roos (2005), Barbour & Xia 1999), and Le Cam (1986).
