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. Poincare, 33 (1997), pp. 651-671]). Birnbaum's theorem for unimodal random variables is extended to the lattice case.