## 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

### Citations:

Zbl 1373.60043; Zbl 1304.52001
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### References:

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