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Let $X,X_1,\ldots,X_n$ be independent identically distributed random variables.
A. Yu. Zaitsev, Approximation of convolutions of probability distributions by infinitely divisible laws under weakened moment restrictions, Zap. Nauchn. Sem. POMI, 194 (1992), 79–90 (in Russian). English translation in: J. Math. Sci. (N. Y.), 75, no. 5 (1995), 1922–1930
1930
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J. E. Littlewood, A. C. Offord, On the number of real roots of a random algebraic equation, Rec. Math. [Mat. Sbornik] N.S., 12 (1943), 277–286
1943
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P. Erdös, On a lemma of Littlewood and Offord, Bull. Amer. Math. Soc., 51 (1945), 898–902
1945
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A. N. Kolmogorov, Two uniform limit theorems for sums of independent random variables, Theory Probab. Appl., 1 (1956), 384–394 (in Russian)
1956
Earlier work this paper cites.
B. A. Rogozin, On the increase of dispersion of sums of independent random variables, Theory Probab. Appl., 6 (1961), 106–108
1961
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C.-G. Esséen, On the Kolmogorov–Rogozin inequality for the concentration function, Z. Wahrscheinlichkeitstheorie und Verw. Gebiete, 5 (1966), 210–216
1966
Earlier work this paper cites.
C.-G. Esséen, On the concentration function of a sum of independent random variables, Z. Wahrscheinlichkeitstheorie und Verw. Gebiete, 9 (1968), 290–308
1968
Earlier work this paper cites.
V. V. Petrov, Sums of independent random variables, Nauka, Moscow, 1972
1972
Earlier work this paper cites.
G. A. Freiman, Foundations of a structural theory of set addition. Kazan’, 1966 (in Russian). English translation in: Translations of Mathematical Monographs, Vol. 37. American Mathematical Society, Providence, R. I., 1973
1973
Earlier work this paper cites.
W. Hengartner, R. Theodorescu, Concentration function. Academic Press, New York, 1973
1973
Earlier work this paper cites.
D. A. Moskvin, G. A. Freiman, A. A. Yudin, Structural theory of set summation, and local limit theorems for independent lattice random variables, Theory Probab. Appl., 19 (1974), 52–62 (in Russian)
1974
Earlier work this paper cites.
A. B. Mukhin, The concentration of the distributions of sums of independent random variables. I; II; III, Izv. Akad. Nauk UzSSR Ser. Fiz.-Mat. Nauk 17, no. 2 (1973), 25–29; 17, no. 6 (1973), 17–23; 17, no. 1 (1976), 15–19 (in Russian)
1976
Earlier work this paper cites.
G. Halász, Estimates for the concentration function of combinatorial number theory and probability, Periodica Mathematica Hungarica, 8 (1977), 197–211
1977
Earlier work this paper cites.
T. V. Arak, Approximation of n n -fold convolutions of distributions, having a nonnegative characteristic function, with accompanying laws, Theory Probab. Appl., 25 (1980), 225–246
1980
Earlier work this paper cites.
T. V. Arak, On the convergence rate in Kolmogorov’s uniform limit theorem. I, Theory Probab. Appl., 26 (1981), 225–245
1981
Earlier work this paper cites.
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, no. 4 (1984), 657–669
1984
Earlier work this paper cites.
1984
Earlier work this paper cites.
A. Yu. Zaitsev, On the uniform approximation of distributions of sums of independent random variables, Theory Probab. Appl., 32, no. 1 (1987), 40–47
1987
Cited alongside, same era.
T. V. Arak, A. Yu. Zaitsev, Uniform limit theorems for sums of independent random variables, Trudy MIAN, 174 (1986), 1–216 (in Russian), English translation in Proc. Steklov Inst. Math., 174 (1988), 1–216
1988
Cited alongside, same era.
A. Yu. Zaitsev, Multidimensional generalized method of triangular functions, Zap. Nauchn. Semin. LOMI, 158 (1987), 81–104 (in Russian). English translation in: J. Soviet Math., 43, no. 6 (1988), 2797–2810
1988
Cited alongside, same era.
A. Yu. Zaitsev, Estimates for the closeness of successive convolutions of multidimensional symmetric distributions, Probab. Theory Rel. Fields, 79, no. 2 (1988), 175–200
1988
Cited alongside, same era.
M. Rudelson, R. Vershynin, Smallest singular value of a random rectangular matrix, Comm. Pure Appl. Math., 62 (2009), 1707–1739
2009
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T. Tao, Van Vu, Inverse Littlewood–Offord theorems and the condition number of random discrete matrices, Ann. of Math., 169, no. 2 (2009), 595–632
2009
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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 (2009), 377–396
2009
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T. Tao, Van Vu, A sharp inverse Littlewood–Offord theorem, Random Structures and Algorithms, 37, no. 4 (2010), 525–539
2010
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Hoi Nguyen, Van Vu, Optimal inverse Littlewood–Offord theorems. Adv. Math. 226 (2011), 5298–5319
2011
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V. Čekanavičius, Approximation by accompanying distributions and asymptotic expansions. I, Litovsk. Mat. Sb., 29, no. 1 (1989), 171–178 (in Russian)
1989
Cited alongside, same era.
A. Yu. Zaitsev, Multivariate version of the second Kolmogorov’s uniform limit theorem, Theory Probab. Appl., 34, no. 1 (1989), 108–128
1989
Cited alongside, same era.
A. Yu. Zaitsev, On the approximation of convolutions of multi-dimensional symmetric distributions by accompaning laws, Zap. Nauchn. Semin. LOMI, 177 (1989), 55–72 (in Russian). English translation in: J. Soviet Math., 61, no. 1 (1992), 1859–1872
1992
Cited alongside, same era.
V. Čekanavičius, Bergström-type asymptotic expansions in the first uniform Kolmogorov’s theorem. In: Probability theory and mathematical statistics, Proceedings of the sixth Vilnius conference (Vilnius, 1993), Grigelionis, B. et al. (eds.) Utrecht: VSP, 1994, pp. 223–238
1994
Cited alongside, same era.
A. Yu. Zaitsev, Certain class of nonuniform estimates in multidimensional limit theorems, Zap. Nauchn. Semin. LOMI, 184 (1990), 92–105 (in Russian). English translation in: J. Math. Sci. (N. Y.), 68, no. 4 (1994), 459–468
1994
Cited alongside, same era.
F. Götze, A. Yu. Zaitsev, Estimates for the rapid decay of concentration functions of n n -fold convolutions, J. Theoret. Probab. 11, no. 3 (1998), 715–731
1998
Cited alongside, same era.
F. Götze, A. Yu. Zaitsev, A multiplicative inequality for concentration functions of n n -fold convolutions, High dimensional probability, v. II (Seattle, WA, 1999), Progr. Probab., v. 47, Birkhäuser Boston, Boston, MA, 2000, pp. 39–47
2000
Cited alongside, same era.
J.-M. Deshouillers, G. A. Freiman, A. A. Yudin, On bounds for the concentration function. II, J. Theoret. Probab., 4 (2001), 813–820
2001
Cited alongside, same era.
Yu. S. Eliseeva, A. Yu. Zaitsev, Estimates for the concentration functions of weighted sums of independent random variables, Theory Probab. Appl., 57 (2012), 767–777
2012
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2012
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2013
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G. A. Freiman, A. A. Yudin, On the measure of large values of the modulus of a trigonometric sum, European J. Combin., 34, no. 8, (2013), 1338–1347
2013
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2013
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S. G. Bobkov, G. P. Chistyakov, Bounds on the maximum of the density for sums of independent random variables, Zap. Nauchn. Semin. POMI, 408 (2012), 62–73 (in Russian), English version: J. Math. Sci. (New York), 199 (2014), 100–106
2014
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2014
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2014
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2014
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2015
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V. Čekanavičius, Approximations Methods in Probability Theory. Universitext: Springer, 2016, 274 p
2016
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