Two–Sided Bounds for PDF’s Maximum of a Sum of Weighted Chi-square Variables

Sergey G. Bobkov, Alexey A. Naumov, Vladimir V. Ulyanov

Research output: Chapter in Book/Report/Conference proceedingConference contribution

Abstract

Two–sided bounds are constructed for a probability density function of a weighted sum of chi-square variables. Both cases of central and non-central chi-square variables are considered. The upper and lower bounds have the same dependence on the parameters of the sum and differ only in absolute constants. The estimates obtained will be useful, in particular, when comparing two Gaussian random elements in a Hilbert space and in multidimensional central limit theorems, including the infinite-dimensional case.

Original languageEnglish (US)
Title of host publicationRecent Developments in Stochastic Methods and Applications - ICSM-5, Selected Contributions
EditorsAlbert N. Shiryaev, Konstantin E. Samouylov, Dmitry V. Kozyrev
PublisherSpringer
Pages178-189
Number of pages12
ISBN (Print)9783030832650
DOIs
StatePublished - 2021
Event5th International Conference on Stochastic Methods, ICSM-5 2020 - Moscow, Russian Federation
Duration: Nov 23 2020Nov 27 2020

Publication series

NameSpringer Proceedings in Mathematics and Statistics
Volume371
ISSN (Print)2194-1009
ISSN (Electronic)2194-1017

Conference

Conference5th International Conference on Stochastic Methods, ICSM-5 2020
Country/TerritoryRussian Federation
CityMoscow
Period11/23/2011/27/20

Bibliographical note

Funding Information:
of Education and Science of the Russian Federation as part of the program of the Moscow Center for Fundamental and Applied Mathematics under the agreement No 075-15-2019-1621. Theorem 2 was proved within the framework of the HSE University Basic Research Program. Research of S. Bobkov was supported by the NSF grant DMS-1855575.

Funding Information:
Theorem 1 has been obtained under support of the Ministry of Education and Science of the Russian Federation as part of the program of the Moscow Center for Fundamental and Applied Mathematics under the agreement No 075-15-2019-1621. Theorem 2 was proved within the framework of the HSE University Basic Research Program. Research of S. Bobkov was supported by the NSF grant DMS-1855575.

Publisher Copyright:
© 2021, The Author(s), under exclusive license to Springer Nature Switzerland AG.

Keywords

  • Chi-square variable
  • Gaussian element
  • Two–sided bounds
  • Weighted sum

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