Geodesic
Arak inequalities for concentration functions and the Littlewood–Offord problem
Teoriâ veroâtnostej i ee primeneniâ, Tome 62 (2017) no. 2, pp. 241-266 Cet article a éte moissonné depuis la source Math-Net.Ru

Voir la notice de l'article

Let $X,X_1,\ldots,X_n$ be independent identically distributed random variables. In this paper we study the behavior of the concentration functions of the weighted sums $\sum_{k=1}^{n}X_ka_k $ depending on the arithmetic structure of the coefficients $a_k$. The results obtained the last 10 years for the concentration functions of weighted sums play an important role in the study of singular numbers of random matrices. Recently, Tao and Vu proposed a so-called inverse principle for the Littlewood–Offord problem. We discuss the relations between this inverse principle and a similar principle for sums of arbitrarily distributed independent random variables formulated by Arak in the 1980s.
Keywords: concentration functions, inequalities, the Littlewood–Offord problem, sums of independent random variables. 
@article{TVP_2017_62_2_a1,
     author = {F. G\"otze and Yu. S. Eliseeva and A. Yu. Zaitsev},
     title = {Arak inequalities for concentration functions and the {Littlewood{\textendash}Offord} problem},
     journal = {Teori\^a vero\^atnostej i ee primeneni\^a},
     pages = {241--266},
     year = {2017},
     volume = {62},
     number = {2},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/TVP_2017_62_2_a1/}
}
TY  - JOUR
AU  - F. Götze
AU  - Yu. S. Eliseeva
AU  - A. Yu. Zaitsev
TI  - Arak inequalities for concentration functions and the Littlewood–Offord problem
JO  - Teoriâ veroâtnostej i ee primeneniâ
PY  - 2017
SP  - 241
EP  - 266
VL  - 62
IS  - 2
UR  - http://geodesic.mathdoc.fr/item/TVP_2017_62_2_a1/
LA  - ru
ID  - TVP_2017_62_2_a1
ER  - 
%0 Journal Article
%A F. Götze
%A Yu. S. Eliseeva
%A A. Yu. Zaitsev
%T Arak inequalities for concentration functions and the Littlewood–Offord problem
%J Teoriâ veroâtnostej i ee primeneniâ
%D 2017
%P 241-266
%V 62
%N 2
%U http://geodesic.mathdoc.fr/item/TVP_2017_62_2_a1/
%G ru
%F TVP_2017_62_2_a1
F. Götze; Yu. S. Eliseeva; A. Yu. Zaitsev. Arak inequalities for concentration functions and the Littlewood–Offord problem. Teoriâ veroâtnostej i ee primeneniâ, Tome 62 (2017) no. 2, pp. 241-266. http://geodesic.mathdoc.fr/item/TVP_2017_62_2_a1/

[1] T. V. Arak, “On the approximation of $n$-fold convolutions of distributions having non-negative characteristic functions with accompanying laws”, Theory Probab. Appl., 25:2 (1981), 221–243 | DOI | MR | Zbl

[2] T. V. Arak, “On the convergence rate in Kolmogorov's uniform limit theorem. I”, Theory Probab. Appl., 26:2 (1982), 219–239 | DOI | MR | Zbl

[3] T. V. Arak, A. Yu. Zaitsev, “Uniform limit theorems for sums of independent random variables”, Proc. Steklov Inst. Math., 174 (1988), 1–222 | MR | Zbl

[4] S. G. Bobkov, G. P. Chistyakov, “Bounds on the maximum of the density for sums of independent random variables”, J. Math. Sci. (N. Y.), 199:2 (2014), 100–106 | DOI | MR | Zbl

[5] V. Čekanavičius, “Approximation with accompanying distributions and asymptotic expansions. I”, Lithuanian Math. J., 29:1 (1989), 75–80 | DOI | MR | Zbl

[6] V. Čekanavičius, “Bergström-type asymptotic expansions in the first uniform Kolmogorov's theorem”, Probability theory and mathematical statistics (Vilnius, 1993), TEV, Vilnius, 1994, 223–238 | MR | Zbl

[7] V. Čekanavičius, Approximations methods in probability theory, Universitext, Springer, Cham, 2016, xii+274 pp. | DOI | MR | Zbl

[8] J.-M. Deshouillers, G. A. Freiman, A. A. Yudin, “On bounds for the concentration function. II”, J. Theoret. Probab., 14:3 (2001), 813–820 | DOI | MR | Zbl

[9] Yu. S. Eliseeva, “Multivariate estimates for the concentration functions of weighted sums of independent, identically distributed random variables”, J. Math. Sci. (N. Y.), 204:1 (2015), 78–89 | DOI | MR | Zbl

[10] Yu. S. Eliseeva, A. Yu. Zaitsev, “Estimates of the concentration functions of weighted sums of independent random variables”, Theory Probab. Appl., 57:4 (2013), 670–678 | DOI | DOI | MR | Zbl

[11] Yu. S. Eliseeva, A. Yu. Zaitsev, “On the Littlewood–Offord problem”, J. Math. Sci. (N. Y.), 214:4 (2016), 467–473 | DOI | MR | Zbl

[12] Yu. S. Eliseeva, F. Götze, A. Yu. Zaitsev, “Estimates for the concentration functions in the Littlewood–Offord problem”, J. Math. Sci. (N. Y.), 206:2 (2015), 146–158 | DOI | Zbl

[13] P. Erdös, “On a lemma of Littlewood and Offord”, Bull. Amer. Math. Soc., 51:12 (1945), 898–902 | DOI | MR | Zbl

[14] C. G. Esseen, “On the Kolmogorov–Rogozin inequality for the concentration function”, Z. Wahrscheinlichkeitstheorie und Verw. Gebiete, 5:3 (1966), 210–216 | DOI | MR | Zbl

[15] C. G. Esseen, “On the concentration function of a sum of independent random variables”, Z. Wahrscheinlichkeitstheorie und Verw. Gebiete, 9:4 (1968), 290–308 | DOI | MR | Zbl

[16] G. A. Freiman, “Foundations of a structural theory of set addition”, Transl. Math. Monogr., 37, Amer. Math. Soc., Providence, R.I., 1973, vii+108 pp. | MR | MR | Zbl | Zbl

[17] G. A. Freiman, A. A. Yudin, “On the measure of large values of the modulus of a trigonometric sum”, European J. Combin., 34:8 (2013), 1338–1347 | DOI | MR | Zbl

[18] O. Friedland, S. Sodin, “Bounds on the concentration function in terms of the Diophantine approximation”, C. R. Math. Acad. Sci. Paris, 345:9 (2007), 513–518 | DOI | MR | Zbl

[19] F. Götze, A. Yu. Zaitsev, “Estimates for the rapid decay of concentration functions of $n$-fold convolutions”, J. Theoret. Probab., 11:3 (1998), 715–731 | DOI | MR | Zbl

[20] F. Götze, A. Yu. Zaitsev, “A multiplicative inequality for concentration functions of $n$-fold convolutions”, High dimensional probability (Seattle, WA, 1999), v. II, Progr. Probab., 47, Birkhäuser Boston, Boston, MA, 2000, 39–47 | MR | Zbl

[21] G. Halász, “Estimates for the concentration function of combinatorial number theory and probability”, Period. Math. Hungar., 8:3 (1977), 197–211 | DOI | MR | Zbl

[22] W. Hengartner, R. Theodorescu, Concentration functions, Probab. Math. Statist., 20, Academic Press, New York–London, 1973, xii+139 pp. | MR | Zbl | Zbl

[23] A. N. Kolmogorov, “Two uniform limit theorems for sums of independent random variables”, Theory Probab. Appl., 1:4 (1956), 384–394 | DOI | MR | Zbl

[24] J. E. Littlewood, A. C. Offord, “On the number of real roots of a random algebraic equation. III”, Matem. sb., 12(54):3 (1943), 277–286 | MR | Zbl

[25] D. A. Moskvin, G. A. Freiman, A. A. Yudin, “Structure theory of set addition and local limit theorems for independent lattice random variables”, Theory Probab. Appl., 19:1 (1974), 53–64 | DOI | MR | Zbl

[26] A. B. Mukhin, “O kontsentratsii raspredelenii summ nezavisimykh sluchainykh velichin. I”, Izv. AN UzSSR. Ser. fiz.-matem. nauk, 1973, no. 2, 25–29 ; II, 1973, No 6, 18–23 ; III, 1976, No 1, 15–19 | Zbl | Zbl

[27] Hoi Nguyen, Van Vu, “Optimal inverse Littlewood–Offord theorems”, Adv. Math., 226:6 (2011), 5298–5319 | DOI | MR | Zbl

[28] Hoi H. Nguyen, Van H. Vu, “Small ball probabilities, inverse theorems, and applications”, Erdős centennial, Bolyai Soc. Math. Stud., 25, János Bolyai Math. Soc., Budapest, 2013, 409–463 | DOI | MR | Zbl

[29] V. V. Petrov, Sums of independent random variables, Ergeb. Math. Grenzgeb., 82, Springer-Verlag, New York–Heidelberg, 1975, x+346 pp. | MR | MR | Zbl | Zbl

[30] B. A. Rogozin, “On the increase of dispersion of sums of independent random variables”, Theory Probab. Appl., 6:1 (1961), 97–99 | DOI | MR | Zbl

[31] M. Rudelson, R. Vershynin, “The Littlewood–Offord problem and invertibility of random matrices”, Adv. Math., 218:2 (2008), 600–633 | DOI | MR | Zbl

[32] M. Rudelson, R. Vershynin, “Smallest singular value of a random rectangular matrix”, Comm. Pure Appl. Math., 62:12 (2009), 1707–1739 | DOI | MR | Zbl

[33] M. Rudelson, R. Vershynin, “Small ball probabilities for linear images of high-dimensional distributions”, Int. Math. Res. Not. IMRN, 2015:19 (2015), 9594–9617 ; arXiv: 1402.4492 | DOI | MR | Zbl

[34] T. Tao, Van. Vu, “John-type theorems for generalized arithmetic progressions and iterated sumsets”, Adv. Math., 219:2 (2008), 428–449 | DOI | MR | Zbl

[35] T. Tao, Van H. Vu, “Inverse Littlewood–Offord theorems and the condition number of random discrete matrices”, Ann. of Math. (2), 169:2 (2009), 595–632 | DOI | MR | Zbl

[36] T. Tao, Van Vu, “From the Littlewood–Offord problem to the circular law: universality of the spectral distribution of random matrices”, Bull. Amer. Math. Soc. (N.S.), 46:3 (2009), 377–396 | DOI | MR | Zbl

[37] T. Tao, Van Vu, “A sharp inverse Littlewood–Offord theorem”, Random Structures Algorithms, 37:4 (2010), 525–539 | DOI | MR | Zbl

[38] R. Vershynin, “Invertibility of symmetric random matrices”, Random Structures Algorithms, 44:2 (2014), 135–182 | DOI | MR | Zbl

[39] A. Yu. Zaitsev, “On the accuracy of approximation of distributions of sums of independent random variables — which are nonzero with a small probability — by means of accompanying laws”, Theory Probab. Appl., 28:4 (1984), 657–669 | DOI | MR | Zbl

[40] A. Yu. Zaitsev, “Multidimensional generalized method of triangular functions”, J. Soviet Math., 43:6 (1988), 2797–2810 | DOI | MR | Zbl

[41] A. Yu. Zaitsev, “On the uniform approximation of distributions of sums of independent random variables”, Theory Probab. Appl., 32:1 (1987), 40–47 | DOI | MR | Zbl

[42] A. Yu. Zaitsev, “Estimates for the closeness of successive convolutions of multidimensional symmetric distributions”, Probab. Theory Related Fields, 79:2 (1988), 175–200 | DOI | MR | Zbl

[43] A. Yu. Zaitsev, “Multidimensional version of the second uniform limit theorem of Kolmogorov”, Theory Probab. Appl., 34:1 (1989), 108–128 | DOI | MR | Zbl

[44] A. Yu. Zaitsev, “Approximation of convolutions of multi-dimensional symmetric distributions by accompanying laws”, J. Soviet Math., 61:1 (1992), 1859–1872 | DOI | MR | Zbl

[45] A. Yu. Zaitsev, “Certain class of nonuniform estimates in multidimensional limit theorems”, J. Math. Sci., 68:4 (1994), 459–468 | DOI | MR | Zbl

[46] A. Yu. Zaitsev, “Approximation of convolutions of probability distributions by infinitely divisible laws under weakened moment restrictions”, J. Math. Sci., 75:5 (1995), 1922–1930 | DOI | MR | Zbl

[47] A. Yu. Zaitsev, “Approximation of a sample by a Poisson point process”, J. Math. Sci. (N. Y.), 128:1 (2005), 2556–2563 | DOI | MR | Zbl

[48] A. Yu. Zaitsev, “A bound for the maximal probability in the Littlewood–Offord problem”, J. Math. Sci. (N. Y.), 219:5 (2016), 743–746 | DOI | MR

[49] A. Yu. Zaitsev, T. V. Arak, “On the rate of convergence in Kolmogorov's second uniform limit theorem”, Theory Probab. Appl., 28:2 (1984), 351–374 | DOI | MR | Zbl