Geodesic
Lower Bounds for Tails of Sums of Independent Symmetric Random Variables
Teoriâ veroâtnostej i ee primeneniâ, Tome 53 (2008) no. 2, pp. 397-403

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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.
Keywords: concentration function, deviation probabilities, symmetric three point convolution, unimodality.
Mots-clés : Bernoulli convolution, Poisson binomial distribution 
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L. Mattner. Lower Bounds for Tails of Sums of Independent Symmetric Random Variables. Teoriâ veroâtnostej i ee primeneniâ, Tome 53 (2008) no. 2, pp. 397-403. http://geodesic.mathdoc.fr/item/TVP_2008_53_2_a16/