G. Christoph, Yu. V. Prokhorov, V. V. Ulyanov, “On distribution of quadratic forms in Gaussian random variables”, Teor. Veroyatnost. i Primenen., 40:2 (1995), 301–312; Theory Probab. Appl., 40:2 (1995), 250

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Teoriya Veroyatnostei i ee Primeneniya, 1995, Volume 40, Issue 2, Pages 301–312 (Mi tvp3478)

This article is cited in 9 scientific papers (total in 9 papers)

On distribution of quadratic forms in Gaussian random variables

G. Christoph^a, Yu. V. Prokhorov^b, V. V. Ulyanov^c

^a Fakultät für Mathematik, Universität Magdeburg, Magdeburg, Germany
^b Steklov Mathematical Institute, Russian Academy of Sciences
^c M. V. Lomonosov Moscow State University, Faculty of Computational Mathematics and Cybernetics

Full-text PDF (1523 kB) Citations (9)

Abstract: Two-sided bounds are constructed for a density function

p (u, a)

$p(u,a)$ of a random variable

| Y - a |^{2}

$|Y-a|^2$ , where

Y

$Y$ is a Gaussian random element in a Hilbert space with zero mean. The estimates are sharp in the sense that starting from large enough

u

$u$ the ratio of upper bound to lower bound equals 8 and does not depend on any parameters of a distribution of

| Y - a |^{2}

$|Y-a|^2$ . The estimates imply two-sided bounds for probabilities

P (| Y - a | > r)

$\mathbf{P}(|Y-a|>r)$

Keywords: Gaussian measure, tail behavior, noncentral

χ^{2}

$\chi^2$ -distribution, distribution of quadratic forms.

Received: 14.04.1995

English version:
Theory of Probability and its Applications, 1995, Volume 40, Issue 2, Pages 250–260
DOI: https://doi.org/10.1137/1140028

Bibliographic databases:

Document Type: Article

Language: Russian

Citation: G. Christoph, Yu. V. Prokhorov, V. V. Ulyanov, “On distribution of quadratic forms in Gaussian random variables”, Teor. Veroyatnost. i Primenen., 40:2 (1995), 301–312; Theory Probab. Appl., 40:2 (1995), 250–260

Citation in format AMSBIB

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\transl

\jour Theory Probab. Appl.

\yr 1995

\vol 40

\issue 2

\pages 250--260

\crossref{https://doi.org/10.1137/1140028}

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Linking options:

https://www.mathnet.ru/eng/tvp3478

https://www.mathnet.ru/eng/tvp/v40/i2/p301

This publication is cited in the following 9 articles:

Gerd Christoph, Vladimir V. Ulyanov, Springer Proceedings in Mathematics & Statistics, 371, Recent Developments in Stochastic Methods and Applications, 2021, 215
Sergey G. Bobkov, Alexey A. Naumov, Vladimir V. Ulyanov, Springer Proceedings in Mathematics & Statistics, 371, Recent Developments in Stochastic Methods and Applications, 2021, 178
Yasunori Fujikoshi, Vladimir V. Ulyanov, SpringerBriefs in Statistics, Non-Asymptotic Analysis of Approximations for Multivariate Statistics, 2020, 81
“Abstracts of talks given at the 3rd International Conference on Stochastic Methods”, Theory Probab. Appl., 64:1 (2019), 124–169
Goetze F. Naumov A. Spokoiny V. Ulyanov V., “Large Ball Probabilities, Gaussian Comparison and Anti-Concentration”, Bernoulli, 25:4A (2019), 2538–2563
Naumov A.A., Spokoiny V.G., Tavyrikov Yu.E., Ulyanov V.V., “Nonasymptotic Estimates For the Closeness of Gaussian Measures on Balls”, Dokl. Math., 98:2 (2018), 490–493
V. V. Ulyanov, “On properties of polynomials in random elements”, Theory Probab. Appl., 60:2 (2016), 325–336
Gerd Christoph, Vladimir V. Ulyanov, Yasunori Fujikoshi, Springer Proceedings in Mathematics & Statistics, 33, Prokhorov and Contemporary Probability Theory, 2013, 239
L. V. Rozovskii, “On Gaussian Measure of Balls in a Hilbert Space”, Theory Probab. Appl., 53:2 (2009), 357–364

Citing articles in Google Scholar: Russian citations, English citations
Related articles in Google Scholar: Russian articles, English articles

Theory of Probability and its Applications

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