##
**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)].

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)].

Reviewer: Viktor Ohanyan (Erevan)

### MSC:

60B12 | Limit theorems for vector-valued random variables (infinite-dimensional case) |

60F05 | Central limit and other weak theorems |

60E05 | Probability distributions: general theory |

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\textit{S. G. Bobkov} et al., Ann. Probab. 48, No. 3, 1202--1219 (2020; Zbl 1476.60008)

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