Skip to main content
Disciplines
Automotive Business IT + Informatics Construction + Real Estate Electrical Engineering + Electronics Energy + Sustainability Insurance + Risk Finance + Banking Management + Leadership Marketing + Sales Mechanical Engineering + Materials
Events
DE
EN
Books
Journals
Topic Page
Marketing
Start single access now
Access for companies
EXTENDED SEARCH
Log in
Top
Published in:
Non-Asymptotic Bounds Read chapter Read first chapter

2020 | OriginalPaper | Chapter

8. Gaussian Comparison and Anti-concentration

Authors : Yasunori Fujikoshi, Vladimir V. Ulyanov

Published in: Non-Asymptotic Analysis of Approximations for Multivariate Statistics

Publisher: Springer Singapore

Log in

Activate our intelligent search to find suitable subject content or patents.

search-config
loading …

Abstract

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 the Pinsker inequality via the Kullback–Leibler divergence. We also establish an anti-concentration bound for a squared norm of a non-centered Gaussian element in a Hilbert space. A number of examples are also provided, motivating the results and its applications to statistical inference and high-dimensional CLT.

Dont have a licence yet? Then find out more about our products and how to get one now:

Springer Professional "Wirtschaft+Technik"

Online-Abonnement

Mit Springer Professional "Wirtschaft+Technik" erhalten Sie Zugriff auf:

  • über 102.000 Bücher
  • über 537 Zeitschriften

aus folgenden Fachgebieten:

  • Automobil + Motoren
  • Bauwesen + Immobilien
  • Business IT + Informatik
  • Elektrotechnik + Elektronik
  • Energie + Nachhaltigkeit
  • Finance + Banking
  • Management + Führung
  • Marketing + Vertrieb
  • Maschinenbau + Werkstoffe
  • Versicherung + Risiko

Jetzt Wissensvorsprung sichern!

inform now

Springer Professional "Technik"

Online-Abonnement

Mit Springer Professional "Technik" erhalten Sie Zugriff auf:

  • über 67.000 Bücher
  • über 390 Zeitschriften

aus folgenden Fachgebieten:

  • Automobil + Motoren
  • Bauwesen + Immobilien
  • Business IT + Informatik
  • Elektrotechnik + Elektronik
  • Energie + Nachhaltigkeit
  • Maschinenbau + Werkstoffe




 

Jetzt Wissensvorsprung sichern!

inform now

Springer Professional "Wirtschaft"

Online-Abonnement

Mit Springer Professional "Wirtschaft" erhalten Sie Zugriff auf:

  • über 67.000 Bücher
  • über 340 Zeitschriften

aus folgenden Fachgebieten:

  • Bauwesen + Immobilien
  • Business IT + Informatik
  • Finance + Banking
  • Management + Führung
  • Marketing + Vertrieb
  • Versicherung + Risiko




Jetzt Wissensvorsprung sichern!

inform now
previous chapter Bootstrap Confidence Sets
next chapter Approximations for Statistics Based on Random Sample Sizes
Literature
1.
Chernozhukov, V., Chetverikov, D., & Kato, K. (2013). Gaussian approximations and multiplier bootstrap for maxima of sums of high-dimensional random vectors. The Annals of Statistics, 41(6), 2786–2819.MathSciNetCrossRef
2.
Chernozhukov, V., Chetverikov, D., & Kato, K. (2015). Comparison and anti-concentration bounds for maxima of Gaussian random vectors. Probability Theory and Related Fields, 162(1–2), 47–70.MathSciNetCrossRef
3.
Chernozhukov, V., Chetverikov, D., & Kato, K. (2017). Central limit theorems and bootstrap in high dimensions. The Annals of Probability, 45(4), 2309–2352.MathSciNetCrossRef
4.
Christoph, G., Prokhorov, Y., & Ulyanov, V. (1996). On distribution of quadratic forms in Gaussian random variables. Theory of Probability and its Applications, 40(2), 250–260.MathSciNetCrossRef
5.
Chung, K. (2001). A course in probability theory (3rd ed.). San Diego: Academic Press Inc.
6.
Götze, G., Naumov, A., Spokoiny, V., & Ulyanov, V. (2019). Large ball probabilities, Gaussian comparison and anti-concentration. Bernoulli, 25(4A), 2538–2563.MathSciNetCrossRef
7.
Markus, A. (1964). The eigen- and singular values of the sum and product of linear operators. Russian Mathematical Surveys, 19, 91–120.CrossRef
8.
Naumov, A. A., Spokoiny, V. G., Tavyrikov, Yu E, & Ulyanov, V. V. (2018). Nonasymptotic estimates for the closeness of Gaussian measures on balls. Doklady Mathematics, 98(2), 490–493.CrossRef
9.
Naumov, A., Spokoiny, V., & Ulyanov, V. (2019). Bootstrap confidence sets for spectral projectors of sample covariance. Probability Theory and Related Fields, 174(3–4), 1091–1132.MathSciNetCrossRef
10.
Panov, M., & Spokoiny, V. (2015). Finite sample Bernstein-von Mises theorem for semiparametric problems. Bayesian Analysis, 10, 665–710.MathSciNetCrossRef
11.
Spokoiny, V., & Zhilova, M. (2015). Bootstrap confidence sets under model misspecification. The Annals of Statistics, 43(6), 2653–2675.MathSciNetCrossRef
12.
Tsybakov, A. (2008). Introduction to nonparametric estimation. New York: Springer.MATH
13.
Ulyanov, V.V. (1995). On Gaussian measure of balls in \(H\). In Frontiers in Pure and Applied Probability, Proceedings of the 4th Russian–Finnish Symposium on Probability Theory and Mathematical Statistics. Moscow: TVP Science.
Metadata
Title
Gaussian Comparison and Anti-concentration
Authors
Yasunori Fujikoshi
Vladimir V. Ulyanov
Copyright Year
2020
Publisher
Springer Singapore
Book
Non-Asymptotic Analysis of Approximations for Multivariate Statistics

Print ISBN: 978-981-13-2615-8

Electronic ISBN: 978-981-13-2616-5

Copyright Year: 2020

DOI
https://doi.org/10.1007/978-981-13-2616-5_8

Premium Partner

    Image Credits