zbMATH — the first resource for mathematics

Berry-Esseen bounds for typical weighted sums. (English) Zbl 1414.60012
Summary: Under correlation-type conditions, we derive upper bounds of order \(\frac{1}{\sqrt{n}}\) for the Kolmogorov distance between the distributions of weighted sums of dependent summands and the normal law.

60F05 Central limit and other weak theorems
60G50 Sums of independent random variables; random walks
Full Text: DOI Euclid
[1] Aistleitner, C. and Berkes, I.: On the central limit theorem for \(f(n_k x)\). Probab. Theory Related Fields146, (2010), no. 1-2, 267–289. · Zbl 1185.60019
[2] Aistleitner, C. and Elsholtz, C.: The central limit theorem for subsequences in probabilistic number theory. Canad. J. Math.64, (2012), no. 6, 1201–1221. · Zbl 1314.60072
[3] Bateman, H.: Higher transcendental functions, Vol. II. McGraw-Hill Book Company, Inc., 1953, 396 pp.
[4] Bobkov, S. G.: On concentration of distributions of random weighted sums. Ann. Probab.31, (2003), no. 1, 195–215. · Zbl 1015.60019
[5] Bobkov, S. G.: On a theorem of V. N. Sudakov on typical distributions. (Russian) J. Math. Sciences (New York)167, (2010), no. 4, 464–473. Translated from: Zap. Nauchn. Semin. POMI, vol. 368, (2009), 59–74. · Zbl 1288.60015
[6] Bobkov, S. G.: Closeness of probability distributions in terms of Fourier-Stieltjes transforms. Russian Math. Surveys, vol. 71, issue 6, (2016), 1021–1079. Translated from: Uspekhi Matem. Nauk, vol. 71, issue 6 (432), (2016), 37–98. · Zbl 1387.60019
[7] Bobkov, S. G., Chistyakov, G. P. and Götze, F.: Gaussian mixtures and normal approximation for V. N. Sudakov’s typical distributions. Zap. Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst. Steklov. (POMI) 457 (2017), Veroyatnost i Statistika. 25, 37–52.
[8] Bobkov, S. G. and Götze, F. On the central limit theorem along subsequences of noncorrelated observations. Theory Probab. Appl.48, (2004), no. 4, 604–621. · Zbl 1063.60023
[9] Bobkov, S. G. and Götze, F.: Concentration inequalities and limit theorems for randomized sums. Probab. Theory Related Fields137, (2007), no. 1–2, 49–81. · Zbl 1111.60014
[10] Fukuyama, K.: A central limit theorem to trigonometric series with bounded gaps. Probab. Theory Related Fields149, (2011), no. 1–2, 139–148. · Zbl 1231.60020
[11] Gaposhkin, V. F.: The rate of approximation to the normal law of the distributions of weighted sums of lacunary series. (Russian) Teor. Verojatnost. i Primenen.13, (1968), 445–461.
[12] Kac, M.: On the distribution of values of sums of the type \(∑ f(2^k t)\). Ann. Math.47 (2), (1946), 33–49. · Zbl 0063.03091
[13] Klartag, B.: A Berry-Esseen type inequality for convex bodies with an unconditional basis. Probab. Theory Related Fields145, (2009), no. 1-2, 1–33. · Zbl 1171.60322
[14] Klartag, B. and Sodin, S.: Variations on the Berry-Esseen theorem. Teor. Veroyatn. Primen.56, (2011), no. 3, 514–533; reprinted in: Theory Probab. Appl.56, (2012), no. 3, 403–419. · Zbl 1285.60018
[15] Matskyavichyus, V. K.: A lower bound for the rate of convergence in the central limit theorem. (Russian) Teor. Veroyatnost. i Primenen.28, (1983), no. 3, 565–569. · Zbl 0511.60024
[16] Meckes, E. S. and Meckes, M. W.: The central limit problem for random vectors with symmetries. J. Theoret. Probab.20, (2007), no. 4, 697–720. · Zbl 1135.60013
[17] Petrov, V. V.: Sums of independent random variables. Translated from the Russian by A. A. Brown. Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 82. Springer–Verlag, New York–Heidelberg, 1975. x+346 pp. · Zbl 0322.60042
[18] Petrov, V. V.: Limit theorems for sums of independent random variables (Russian), Nauka, Moscow, 1987. 318 pp.
[19] Salem, R. and Zygmund, A.: On lacunary trigonometric systems. Proc. Nat. Acad. Sci. USA33, (1947), 333–338. · Zbl 0029.11902
[20] Salem, R. and Zygmund, A.: On lacunary trigonometric series. II. Proc. Nat. Acad. Sci. USA34, (1948), 54–62. · Zbl 0029.35601
[21] Sudakov, V. N.: Typical distributions of linear functionals in finite-dimensional spaces of high dimension. (Russian) Soviet Math. Dokl.19, (1978), 1578–1582; translation in: Dokl. Akad. Nauk SSSR243, (1978), no. 6, 1402–1405. · Zbl 0416.60005
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.