  The approach of Kleitman (1970) and Kanter (1976) to multivariate
concentration function inequalities is generalized in order to obtain for
deviation probabilities of sums of independent symmetric random variables a
lower bound depending only on deviation probabilities of the terms of the sum.
This bound is optimal up to discretization effects, improves on a result of
Nagaev (2001), and complements the comparison theorems of Birnbaum (1948) and
Pruss (1997). Birnbaum's theorem for unimodal random variables is extended to
the lattice case.
