Some Approximation Problems in Statistics and Probability

Prokhorov, Yuri V.; Ulyanov, V. V.

doi:10.1007/978-3-642-36068-8_11

Cited by 24 publications

(19 citation statements)

References 31 publications

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“…Such classes naturally appear in various statistical problems; see e.g. Prokhorov and Ulyanov (2013) or our previous application examples. Thus, similar to the inequalities (1.9) we need to get sharp bounds for the 10 F. Götze et al probability IP (x < X − a 2 < x + ε) for the Gaussian element X in a Hilbert space H .…”

Section: Central Limit Theorem In Finite-and Infinite-dimensional Spacesmentioning

confidence: 98%

Large ball probabilities, Gaussian comparison and anti-concentration

Götze¹,

Naumov²,

Spokoiny³

et al. 2019

Bernoulli

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We derive tight non-asymptotic bounds for the Kolmogorov distance between the probabilities of two Gaussian elements to hit a ball in a Hilbert space. The key property of these bounds is that they are dimension-free and depend on the nuclear (Schatten-one) norm of the difference between the covariance operators of the elements and on the norm of the mean shift. The obtained bounds significantly improve the bound based on Pinsker's inequality via the Kullback-Leibler divergence. We also establish an anti-concentration bound for a squared norm of a non-centered Gaussian element in Hilbert space. The paper presents a number of examples motivating our results and applications of the obtained bounds to statistical inference and to high-dimensional CLT.

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Section: Central Limit Theorem In Finite-and Infinite-dimensional Spacesmentioning

confidence: 98%

Large ball probabilities, Gaussian comparison and anti-concentration

Götze¹,

Naumov²,

Spokoiny³

et al. 2019

Bernoulli

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show abstract

“…. , be independent random vectors with zero mean, nondegenerate covariance matrix (see (7)) and finite absolute moment of order ≥ 3. Then there are constants 1 ( , ) and 2 ( , ) such that for any | | ≤ and…”

Section: Notation and Auxiliary Resultsmentioning

confidence: 99%

A multivariate CLT for <> weighted sums with rate of convergence of order O(1/n)

A.¹,

Ulyanov²

2021

Preprint

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“…See the upper bounds for ∆ (a) N with s = 12 and d = ∞ in Ulyanov and Götze (2011), where the dependence on the eigenvalues of C is given in the upper bound in an explicit form which coincides with that in the lower bound. See also the review of recent results for "almost" quadratic forms in Prokhorov and Ulyanov (2013).…”

Section: Resultsmentioning

confidence: 99%

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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Some Approximation Problems in Statistics and Probability

Cited by 24 publications

References 31 publications

Large ball probabilities, Gaussian comparison and anti-concentration

Large ball probabilities, Gaussian comparison and anti-concentration

A multivariate CLT for <> weighted sums with rate of convergence of order O(1/n)

Explicit rates of approximation in the CLT for quadratic forms

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