2014
DOI: 10.1214/13-aop839
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Explicit rates of approximation in the CLT for quadratic forms

Abstract: Let $X,X_1,X_2,\ldots$ be i.i.d. ${\mathbb{R}}^d$-valued real random vectors. Assume that ${\mathbf{E}X=0}$, $\operatorname {cov}X=\mathbb{C}$, $\mathbf{E}\Vert X\Vert^2=\sigma ^2$ and that $X$ is not concentrated in a proper subspace of $\mathbb{R}^d$. Let $G$ be a mean zero Gaussian random vector with the same covariance operator as that of $X$. We study the distributions of nondegenerate quadratic forms $\mathbb{Q}[S_N]$ of the normalized sums ${S_N=N^{-1/2}(X_1+\cdots+X_N)}$ and show that, without any addi… Show more

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Cited by 26 publications
(22 citation statements)
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“…), where 2 = −1 ( see (8)), it follows that = . Using Holder's inequality, we obtain that for the characteristic function of the random vector at ≤ − 1 the following inequality holds…”
Section: Notation and Auxiliary Resultsmentioning
confidence: 99%
See 1 more Smart Citation
“…), where 2 = −1 ( see (8)), it follows that = . Using Holder's inequality, we obtain that for the characteristic function of the random vector at ≤ − 1 the following inequality holds…”
Section: Notation and Auxiliary Resultsmentioning
confidence: 99%
“…, where the matrix is defined in (8). Also, taking into account that for any ≥ 4 from (16) it follows…”
Section: Proof Of the Main Theoremmentioning
confidence: 99%
“…For Euclidean balls centered at 0, it is known that (cf. Götze and Zaitsev (2014)) the dependence on n may be improved from 1/ √ n to 1/n for d ≥ 5, which is in general the smallest possible dimension for such an improvement. See also Esseen (1945), Bentkus and Götze (1997), Götze and Ulyanov (2003), Bogatyrev, Götze and Ulyanov (2006) and Prokhorov and Ulyanov (2013) for earlier and related results.…”
Section: Literature On Multivariate Normal Approximationsmentioning
confidence: 99%
“…The notation O r emphasizes that the hidden constant depends on r. To our knowledge, there are no similar bounds for d K (Law (T J ) , χ 2 (r − 1)) in the literature. Fortunately, one can derive even better rates of convergence than O r (n −1+1/r ) using recent advances in CLT for quadratic forms [17,Theorem 1.2] in addition to the classical result [9, Chapter 7, Theorem 1]. We provide the details in Proposition 2.3 below.…”
Section: Related Work 21 Test Statistics Based On N Rankingsmentioning
confidence: 99%
“…The main obstacle in the analysis of T φ λ , λ = 1, is that, in contrast to Pearson's statistic, it is not a quadratic form anymore, and consequently, the results of Götze [15, Theorem 1.5] or of Götze and Zaitsev [17,Theorem 1.2] are not applicable anymore. Due to Taylor's expansion, the phi-divergence test statistic T φ can be represented in the form…”
Section: Phi-divergence Test Statisticmentioning
confidence: 99%