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 Cet article a éte moissonné depuis la source Math-Net.Ru

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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/

[1] Berger E., “Comparing sums of independent bounded random variables and sums of Bernoulli random variables”, Statist. Probab., 34:3 (1997), 251–258 | DOI | MR | Zbl

[2] Bickel P. J., Lehmann E. L., “Descriptive statistics for nonparametric models. III. Dispersion”, Ann. Statist., 4:6 (1976), 1139–1158 | DOI | MR | Zbl

[3] Birnbaum Z. W., “On random variables with comparable peakedness”, Ann. Math. Statist., 19 (1948), 76–81 | DOI | MR | Zbl

[4] Dharmadhikari S., Joag-Dev K., Unimodality, Convexity, and Applications, Academic Press, Boston, 1988, 278 pp. | MR | Zbl

[5] Erdős P., “On a lemma of Littlewood and Offord”, Bull. Amer. Math. Soc., 51 (1945), 898–902 | DOI | MR | Zbl

[6] Jones L., “On the distribution of sums of vectors”, SIAM J. Appl. Math., 34:1 (1978), 1–6 | DOI | MR | Zbl

[7] Le Cam, Asymptotic Methods in Statistical Decision Theory, Springer-Verlag, New York, 1986, 742 pp. | MR

[8] Kanter M., “Probability inequalities for convex sets and multidimensional concentration functions”, J. Multivariate Anal., 6:2 (1976), 222–236 | DOI | MR | Zbl

[9] Kleitman D. J., “On a lemma of Littlewood and Offord on the distributions of linear combinations of vectors”, Adv. in Math., 5 (1970), 155–157 | DOI | MR | Zbl

[10] Mattner L., Roos B., “A shorter proof of Kanter's Bessel function concentration bound”, Probab. Theory Related Fields, 139:1–2 (2007), 191–205 | DOI | MR | Zbl

[11] Nagaev S. V., “Nizhnie otsenki dlya veroyatnostei bolshikh uklonenii summ nezavisimykh sluchainykh velichin”, Teoriya veroyatn. i ee primen., 46:4 (2001), 785–792 | MR | Zbl

[12] Pruss A. R., “Comparisons between tail probabilities of sums of independent symmetric random variables”, Ann. Inst. Henri Poincaré, 33:5 (1997), 651–671 | DOI | MR | Zbl

[13] Rudin W., Functional Analysis, 2nd ed., McGraw-Hill, New York, 1991, 424 pp. | MR | Zbl

[14] Shaked M., Shantikumar J. G., Stochastic Orders and their Applications, Academic Press, Boston, 1994, 545 pp. | MR | Zbl

[15] Sherman S., “A theorem on convex sets with applications”, Ann. Math. Statist., 26 (1955), 763–767 | DOI | MR | Zbl