Lower Bounds for Tails of Sums of Independent Symmetric Random Variables

The approach of Kleitman [Adv. in Math., 5 (1970), pp. 155–157] and Kanter [J. Multivariate Anal., 6 (1976), pp. 222–236] 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 [Theory Probab. Appl., 46 (2002), pp. 728–735], and complements the comparison theorems of Birnbaum [Ann. Math. Statist., 19 (1948), pp. 76–81] and Pruss [Ann. Inst. H. Poincaré, 33 (1997), pp. 651–671]). Birnbaum's theorem for unimodal random variables is extended to the lattice case.