Geodesic
Bound for the maximal probability in the Littlewood–Offord problem
Zapiski Nauchnykh Seminarov POMI, Probability and statistics. Part 22, Tome 441 (2015), pp. 204-209 Cet article a éte moissonné depuis la source Math-Net.Ru

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The paper deals with studying a connection of the Littlewood–Offord problem with estimating the concentration functions of some symmetric infinitely divisible distributions. It is shown that the values at zero of the concentration functions of weighted sums of i.i.d. random variables may be estimated by the values at zero of the concentration functions of symmetric infinitely divisible distributions with the Lévy spectral measures which are multiples of the sum of delta-measures at $\pm$weights involved in constructing the weighted sums. 
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A. Yu. Zaitsev. Bound for the maximal probability in the Littlewood–Offord problem. Zapiski Nauchnykh Seminarov POMI, Probability and statistics. Part 22, Tome 441 (2015), pp. 204-209. http://geodesic.mathdoc.fr/item/ZNSL_2015_441_a12/

[1] T. V. Arak, “O sblizhenii $n$-kratnykh svertok raspredelenii, imeyuschikh neotritsatelnuyu kharakteristicheskuyu funktsiyu, s soprovozhdayuschimi zakonami”, Teoriya veroyatn. i ee primen., 25:2 (1980), 225–246 | MR | Zbl

[2] T. V. Arak, “O skorosti skhodimosti v ravnomernoi predelnoi teoreme Kolmogorova. I”, Teoriya veroyatn. i ee primen., 26:2 (1981), 225–245 | MR | Zbl

[3] T. V. Arak, A. Yu. Zaitsev, Ravnomernye predelnye teoremy dlya summ nezavisimykh sluchainykh velichin, Tr. MIAN SSSR, 174, 1986, 214 pp. | MR | Zbl

[4] Yu. S. Eliseeva, “Mnogomernye otsenki funktsii kontsentratsii vzveshennykh summ nezavisimykh odinakovo raspredelennykh sluchainykh velichin”, Zap. nauchn. semin. POMI, 412, 2013, 121–137 | MR

[5] Yu. S. Eliseeva, F. Gëttse, A. Yu. Zaitsev, “Otsenki funktsii kontsentratsii v probleme Littlvuda–Offorda”, Zap. nauchn. semin. POMI, 420, 2013, 50–69

[6] Yu. S. Eliseeva, F. Götze, A. Yu. Zaitsev, Arak inequalities for concentration functions and the Littlewood–Offord problem, 2015, arXiv: 1506.09034

[7] Yu. S. Eliseeva, A. Yu. Zaitsev, “Otsenki funktsii kontsentratsii vzveshennykh summ nezavisimykh odinakovo raspredelennykh sluchainykh velichin”, Teoriya veroyatn. i ee primen., 57:4 (2012), 768–777 | DOI | Zbl

[8] Yu. S. Eliseeva, A. Yu. Zaitsev, “O probleme Littlvuda–Offorda”, Zap. nauchn. semin. POMI, 431, 2014, 72–81

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

[10] V. Khengartner, R. Teodoresku, Funktsii kontsentratsii, Nauka, M., 1980 | MR

[11] J. E. Littlewood, A. C. Offord, “On the number of real roots of a random algebraic equation”, Rec. Math. [Mat. Sbornik] N.S., 12(54):3 (1943), 277–286 | MR | Zbl

[12] H. Nguyen, V. Vu, “Optimal inverse Littlewood–Offord theorems”, Adv. Math., 226 (2011), 5298–5319 | DOI | MR | Zbl

[13] H. Nguyen, V. Vu, “Small ball probability, inverse theorems, and applications”, Erdös Centennial Proceeding, eds. L. Lovász et. al., Springer, 2013, 409–463 | DOI | MR | Zbl

[14] M. Rudelson, R. Vershynin, “The Littlewood–Offord problem and invertibility of random matrices”, Adv. Math., 218 (2008), 600–633 | DOI | MR | Zbl

[15] M. Rudelson, R. Vershynin, “The smallest singular value of a random rectangular matrix”, Comm. Pure Appl. Math., 62 (2009), 1707–1739 | DOI | MR | Zbl

[16] T. Tao, V. Vu, “Inverse Littlewood–Offord theorems and the condition number of random discrete matrices”, Ann. Math., 169 (2009), 595–632 | DOI | MR | Zbl

[17] T. Tao, V. Vu, “From the Littlewood–Offord problem to the circular law: universality of the spectral distribution of random matrices”, Bull. Amer. Math. Soc., 46 (2009), 377–396 | DOI | MR | Zbl

[18] R. Vershynin, “Invertibility of symmetric random matrices”, Random Structures Algorithms, 44 (2014), 135–182 | DOI | MR | Zbl