Non-uniform bounds for short asymptotic expansions in the CLT for balls in a Hilbert space

Bogatyrev, S. A.; Götze, Friedrich; Ulyanov, V. V.

doi:10.1016/j.jmva.2006.01.010

Cited by 5 publications

(5 citation statements)

References 14 publications

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“…Similar bounds for the rate of infinitely divisible approximations were obtained by Bentkus, Götze and Zaitsev (1997). Among recent publications, we should mention the papers of Nagaev andChebotarev (1999, 2005) (d ≥ 13, providing a more precise dependence of constants on the eigenvalues of C) and Bogatyrev, Götze and Ulyanov (2006) (nonuniform bounds for d ≥ 12); see also Götze and Ulyanov (2000). The proofs of bounds of order O(N −1 ) are based on discretization (i.e., a reduction to lattice valued random vectors) and the symmetrization techniques mentioned above.…”

supporting

confidence: 63%

“…For balls, Q = I d . In the papers mentioned above, the authors have used the aproach of BG (1997a) and obtained bounds with constants depending on s ≤ d largest eigenvalues σ 2 1 ≥ σ 2 2 ≥ · · · ≥ σ 2 s of the covariance operator C; see Nagaev andChebotarev (1999, 2005), with d ≥ s = 13, and Götze and Ulyanov (2000), and Bogatyrev, Götze and Ulyanov (2006), with d ≥ s = 12. It should be mentioned, that, in a particular case, where Q = I d and d ≥ 12, these results may be sharper than (2.22), for some covariance operators C. The lower bounds for ∆ (a) N under different conditions on a and L(X) are given in Götze and Ulyanov (2000).…”

Section: Resultsmentioning

confidence: 99%

“…s of the covariance operator C; see Nagaev andChebotarev (1999, 2005), with d ≥ s = 13, and Götze and Ulyanov (2000), and Bogatyrev, Götze and Ulyanov (2006), with d ≥ s = 12. It should be mentioned, that, in a particular case, where Q = I d and d ≥ 12, these results may be sharper than (2.22), for some covariance operators C.…”

Section: Resultsmentioning

confidence: 99%

“…providing a more precise dependence of constants on the eigenvalues of C) and Bogatyrev, Götze and Ulyanov (2006) (nonuniform bounds for d ≥ 12); see also Götze and Ulyanov (2000). The proofs of bounds of order O(N −1 ) are based on discretization (i.e., a reduction to lattice valued random vectors) and the symmetrization techniques mentioned above.…”

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confidence: 99%

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Explicit rates of approximation in the CLT for quadratic forms

Götze¹,

Зайцев²

2014

Ann. Probab.

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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 additional conditions, \[\Delta_N\stackrel{\mathrm{def}}{=}\sup_x\bigl |\mathbf{P}\bigl\{\mathbb{Q}[S_N]\leq x\bigr\}-\mathbf{P}\bigl\{\mathbb{Q}[G]\leq x\bigr\}\bigr|={\mathcal{O}}\bigl(N^{-1}\bigr),\] provided that $d\geq5$ and the fourth moment of $X$ exists. Furthermore, we provide explicit bounds of order ${\mathcal{O}}(N^{-1})$ for $\Delta_N$ for the rate of approximation by short asymptotic expansions and for the concentration functions of the random variables $\mathbb{Q}[S_N+a]$, $a\in{\mathbb{R}}^d$. The order of the bound is optimal. It extends previous results of Bentkus and G\"{o}tze [Probab. Theory Related Fields 109 (1997a) 367-416] (for ${d\ge9}$) to the case $d\ge5$, which is the smallest possible dimension for such a bound. Moreover, we show that, in the finite dimensional case and for isometric $\mathbb{Q}$, the implied constant in ${\mathcal{O}}(N^{-1})$ has the form $c_d\sigma ^d(\det\mathbb{C})^{-1/2}\mathbf {E}\|\mathbb{C}^{-1/2}X\|^4$ with some $c_d$ depending on $d$ only. This answers a long standing question about optimal rates in the central limit theorem for quadratic forms starting with a seminal paper by Ess\'{e}en [Acta Math. 77 (1945) 1-125].Comment: Published in at http://dx.doi.org/10.1214/13-AOP839 the Annals of Probability (http://www.imstat.org/aop/) by the Institute of Mathematical Statistics (http://www.imstat.org

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supporting

confidence: 63%

Section: Resultsmentioning

confidence: 99%

Section: Resultsmentioning

confidence: 99%

mentioning

confidence: 99%

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Explicit rates of approximation in the CLT for quadratic forms

Götze¹,

Зайцев²

2014

Ann. Probab.

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“…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. We do not know any results with explicit dependence on d and such improved dependence on n.…”

Section: Literature On Multivariate Normal Approximationsmentioning

confidence: 91%

High-dimensional central limit theorems by Stein’s method

Fang

Koike

2021

Ann. Appl. Probab.

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We obtain explicit error bounds for the d-dimensional normal approximation on hyperrectangles for a random vector that has a Stein kernel, or admits an exchangeable pair coupling, or is a nonlinear statistic of independent random variables or a sum of n locally dependent random vectors. We assume the approximating normal distribution has a nonsingular covariance matrix. The error bounds vanish even when the dimension d is much larger than the sample size n. We prove our main results using the approach of Götze (1991) in Stein's method, together with modifications of an estimate of Anderson, Hall and Titterington (1998) and a smoothing inequality of Bhattacharya and Rao (1976). For sums of n independent and identically distributed isotropic random vectors having a log-concave density, we obtain an error bound that is optimal up to a log n factor. We also discuss an application to multiple Wiener-Itô integrals.

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Some Approximation Problems in Statistics and Probability

Prokhorov

Ulyanov

2013

Springer Proceedings in Mathematics &Amp; Statistics

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Non-uniform bounds for short asymptotic expansions in the CLT for balls in a Hilbert space

Cited by 5 publications

References 14 publications

Explicit rates of approximation in the CLT for quadratic forms

Explicit rates of approximation in the CLT for quadratic forms

High-dimensional central limit theorems by Stein’s method

Some Approximation Problems in Statistics and Probability

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