×

Normal approximation for weighted sums under a second-order correlation condition. (English) Zbl 1476.60008

Let \(X = (X_1, \ldots, X_n)\) be an isotropic random vector in \(\mathbb{R}^n\), that is, with uncorrelated components having mean zero and variance one. Consider the following weighted sums: \(S_{\theta} = \theta_1 X_1 + \ldots + \theta_n X_n\), \(\theta = (\theta_1, \ldots, \theta_n)\), \(\theta_1^2 + \ldots + \theta_n^2= 1\), where the coefficients \(\theta_k\) are taken from the unit sphere \(\mathbf{S}^{n-1}\) in \(\mathbb{R}^n\). Obviously, \(E S_{\theta}=0\) and Var\((S_{\theta}) = 1\). The central limit problem is to determine natural conditions on \(X\) and \(\theta\) which ensure that the random variable \(S_{\theta}\) are nearly standard normal. In this case, one would also like to explore the rate of normal approximation in the Kolmogorov distance \(\rho(F_{\theta}, \Phi) = \sup_x|F_{\theta}(x) - \Phi(x)|\), where \(\Phi(x)\) is the standard normal distribution function. There are interesting results in the case when the components \(X_k\) are independent. One of general variants of the central limit theorem asserts that \(\rho(F_{\theta}, \Phi)\) will be small, as long as \(X_k\) are identically distributed (the i.i.d. case), while \(\max_k |\theta_k|\) is small. Moreover, under the third moment condition this property may be quantified by virtue of the Berry-Essen bound \(\rho(F_{\theta}, \Phi)\le c\, \sum_{k=1}^n|\theta_k|^3\, E|X_k|^3\) (note that this inequality extends to the non-i.i.d. case). The aim of this paper is to extend these results under a suitable correlation-type condition (and thus for some class of dependent \(X_k\)) to isotropic random vectors with a similar \(\frac{1}{n}\)-rate modulo a logarithmic factor. A random vector \(X\) satisfies a second-order correlation condition with constant \(\Lambda\), if for any collection \(a_{ij}\in\mathbb{R}^n\), Var\(\left(\sum_{i,j=1}^n a_{ij} X_i X_j\right) \le \Lambda \sum_{i,j=1}^na_{ij}^2\). An optimal value \(\Lambda= \Lambda(X)\) is finite as long as \(|X|\) has a finite fourth moment.
Theorem 1. Let \(X\) be an isotropic random vector in \(\mathbb{R}^n\) with a symmetric distribution and a finite constant \(\Lambda= \Lambda(X)\). Then \(E_{\theta}\rho(F_{\theta}, \Phi)\le \frac{c \log n}{n} \Lambda\).
The paper is organized as follows. Section 2 contains brief discussion of general properties of \(\Lambda\) and related functionals. There exist several natural classes of probability distributions for which a bound on the parameter \(\Lambda\) can be obtained. Some of them are considered in Section 3. Some results about the second-order concentration on the sphere are described in Section 4, then the results of Section 4 are used in Section 5 to explore the concentration of characteristic functions of \(S_{\theta}\) with respect to the variable \(\theta\). In Section 6 the authors finalize the proof of Theorem 1. The proof of Theorem 1 is based on results for spherical concentration, which have been recently developed in [the first author et al., Commun. Contemp. Math. 19, No. 5, Article ID 1650058, 20 p. (2017; Zbl 1373.60043)]. Section 7 considers the log-concave case. Section 8 refines the relationship between the central limit theorem and the thin-shell problem. Section 9 considers some technical result. Section 10 considers a short overview on the results related to Theorem 1. Some account can also be found in the book [S. Brazitikos et al., Geometry of isotropic convex bodies. Providence, RI: American Mathematical Society (AMS) (2014; Zbl 1304.52001)].

MSC:

60B12 Limit theorems for vector-valued random variables (infinite-dimensional case)
60F05 Central limit and other weak theorems
60E05 Probability distributions: general theory
PDF BibTeX XML Cite
Full Text: DOI arXiv Euclid

References:

[1] Anttila, M., Ball, K. and Perissinaki, I. (2003). The central limit problem for convex bodies. Trans. Amer. Math. Soc. 355 4723-4735. · Zbl 1033.52003
[2] Bobkov, S. G. (2003). On concentration of distributions of random weighted sums. Ann. Probab. 31 195-215. · Zbl 1015.60019
[3] Bobkov, S. G. (2003). Concentration of distributions of the weighted sums with Bernoullian coefficients. In Geometric Aspects of Functional Analysis. Lecture Notes in Math. 1807 27-36. Springer, Berlin. · Zbl 1033.60022
[4] Bobkov, S. G. (2004). Concentration of normalized sums and a central limit theorem for noncorrelated random variables. Ann. Probab. 32 2884-2907. · Zbl 1065.60006
[5] Bobkov, S. G. (2009). On a theorem of V. N. Sudakov on typical distributions. Zap. Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst. Steklov. (POMI) 368 59-74, 283.
[6] Bobkov, S. G., Chistyakov, G. P. and Götze, F. Concentration and Gaussian approximation for randomized sums. In preparation.
[7] Bobkov, S. G., Chistyakov, G. P. and Götze, F. (2017). Second-order concentration on the sphere. Commun. Contemp. Math. 19 1650058, 20. · Zbl 1373.60043
[8] Bobkov, S. G., Chistyakov, G. P. and Götze, F. (2017). Gaussian mixtures and normal approximation for V. N. Sudakov’s typical distributions. Zap. Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst. Steklov. (POMI) 457 37-52. · Zbl 1478.60072
[9] Bobkov, S. G., Chistyakov, G. P. and Götze, F. (2018). Berry-Esseen bounds for typical weighted sums. Electron. J. Probab. 23 Paper No. 92, 22. · Zbl 1414.60012
[10] Bobkov, S. G. and Götze, F. (2007). Concentration inequalities and limit theorems for randomized sums. Probab. Theory Related Fields 137 49-81. · Zbl 1111.60014
[11] Bobkov, S. G. and Koldobsky, A. (2003). On the central limit property of convex bodies. In Geometric Aspects of Functional Analysis. Lecture Notes in Math. 1807 44-52. Springer, Berlin. · Zbl 1039.52003
[12] Borell, C. (1974). Convex measures on locally convex spaces. Ark. Mat. 12 239-252. · Zbl 0297.60004
[13] Brazitikos, S., Giannopoulos, A., Valettas, P. and Vritsiou, B.-H. (2014). Geometry of Isotropic Convex Bodies. Mathematical Surveys and Monographs 196. Amer. Math. Soc., Providence, RI. · Zbl 1304.52001
[14] Diaconis, P. and Freedman, D. (1984). Asymptotics of graphical projection pursuit. Ann. Statist. 12 793-815. · Zbl 0559.62002
[15] Eldan, R. (2013). Thin shell implies spectral gap up to polylog via a stochastic localization scheme. Geom. Funct. Anal. 23 532-569. · Zbl 1277.52013
[16] Eldan, R. and Klartag, B. (2008). Pointwise estimates for marginals of convex bodies. J. Funct. Anal. 254 2275-2293. · Zbl 1144.60006
[17] Guédon, O. and Milman, E. (2011). Interpolating thin-shell and sharp large-deviation estimates for isotropic log-concave measures. Geom. Funct. Anal. 21 1043-1068. · Zbl 1242.60012
[18] Klartag, B. (2007). A central limit theorem for convex sets. Invent. Math. 168 91-131. · Zbl 1144.60021
[19] Klartag, B. (2007). Power-law estimates for the central limit theorem for convex sets. J. Funct. Anal. 245 284-310. · Zbl 1140.52004
[20] Klartag, B. (2009). A Berry-Esseen type inequality for convex bodies with an unconditional basis. Probab. Theory Related Fields 145 1-33. · Zbl 1171.60322
[21] Klartag, B. and Sodin, S. (2011). Variations on the Berry-Esseen theorem. Teor. Veroyatn. Primen. 56 514-533. · Zbl 1285.60018
[22] Ledoux, M. (1999). Concentration of measure and logarithmic Sobolev inequalities. In Séminaire de Probabilités, XXXIII. Lecture Notes in Math. 1709 120-216. Springer, Berlin. · Zbl 0957.60016
[23] Lee, Y. T. and Vempala, S. S. (2017). Eldan’s stochastic localization and the KLS hyperplane conjecture: An improved lower bound for expansion. In 58th Annual IEEE Symposium on Foundations of Computer Science—FOCS 2017 998-1007. IEEE Computer Soc., Los Alamitos, CA.
[24] Meckes, E. S. and Meckes, M. W. (2007). The central limit problem for random vectors with symmetries. J. Theoret. Probab. 20 697-720. · Zbl 1135.60013
[25] Meckes, M. W. (2009). Gaussian marginals of convex bodies with symmetries. Beitr. Algebra Geom. 50 101-118. · Zbl 1162.60006
[26] Milman, V. D. and Schechtman, G. (1986). Asymptotic Theory of Finite-Dimensional Normed Spaces. Lecture Notes in Math. 1200. Springer, Berlin.
[27] Nagaev, S. V. (1982). On the distribution of linear functionals in finite-dimensional spaces of large dimension. Dokl. Akad. Nauk SSSR 263 295-297.
[28] Petrov, V. V. (1975). Sums of Independent Random Variables. Springer, New York. Translated from the Russian by A. A. Brown. Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 82. · Zbl 0322.60043
[29] Petrov, V. V. (1987). Limit Theorems for Sums of Independent Random Variables (Russian). Probability Theory and Mathematical Statistics. Nauka, Moscow.
[30] Sudakov, V. N. (1978). Typical distributions of linear functionals in finite-dimensional spaces of high dimension. Dokl. Akad. Nauk SSSR 243 1402-1405.
[31] von Weizsäcker, H. · Zbl 0868.60009
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.