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

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

doi:10.1007/978-3-030-83266-7_13

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

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

School of Mathematics

Research output: Chapter in Book/Report/Conference proceeding › Conference contribution

1 Scopus citations

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 language	English (US)
Title of host publication	Recent Developments in Stochastic Methods and Applications - ICSM-5, Selected Contributions
Editors	Albert N. Shiryaev, Konstantin E. Samouylov, Dmitry V. Kozyrev
Publisher	Springer
Pages	178-189
Number of pages	12
ISBN (Print)	9783030832650
DOIs	https://doi.org/10.1007/978-3-030-83266-7_13
State	Published - Aug 3 2021
Event	5th International Conference on Stochastic Methods, ICSM-5 2020 - Moscow, Russian Federation Duration: Nov 23 2020 → Nov 27 2020

Publication series

Name	Springer Proceedings in Mathematics and Statistics
Volume	371
ISSN (Print)	2194-1009
ISSN (Electronic)	2194-1017

Conference

Conference	5th International Conference on Stochastic Methods, ICSM-5 2020
Country/Territory	Russian Federation
City	Moscow
Period	11/23/20 → 11/27/20

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

Publisher link

10.1007/978-3-030-83266-7_13

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Bobkov, S. G., Naumov, A. A., & Ulyanov, V. V. (2021). Two–Sided Bounds for PDF’s Maximum of a Sum of Weighted Chi-square Variables. In A. N. Shiryaev, K. E. Samouylov, & D. V. Kozyrev (Eds.), Recent Developments in Stochastic Methods and Applications - ICSM-5, Selected Contributions (pp. 178-189). (Springer Proceedings in Mathematics and Statistics; Vol. 371). Springer. https://doi.org/10.1007/978-3-030-83266-7_13

Two–Sided Bounds for PDF’s Maximum of a Sum of Weighted Chi-square Variables. / Bobkov, Sergey G.; Naumov, Alexey A.; Ulyanov, Vladimir V.
Recent Developments in Stochastic Methods and Applications - ICSM-5, Selected Contributions. ed. / Albert N. Shiryaev; Konstantin E. Samouylov; Dmitry V. Kozyrev. Springer, 2021. p. 178-189 (Springer Proceedings in Mathematics and Statistics; Vol. 371).

Research output: Chapter in Book/Report/Conference proceeding › Conference contribution

Bobkov, SG, Naumov, AA & Ulyanov, VV 2021, Two–Sided Bounds for PDF’s Maximum of a Sum of Weighted Chi-square Variables. in AN Shiryaev, KE Samouylov & DV Kozyrev (eds), Recent Developments in Stochastic Methods and Applications - ICSM-5, Selected Contributions. Springer Proceedings in Mathematics and Statistics, vol. 371, Springer, pp. 178-189, 5th International Conference on Stochastic Methods, ICSM-5 2020, Moscow, Russian Federation, 11/23/20. https://doi.org/10.1007/978-3-030-83266-7_13

Bobkov SG, Naumov AA, Ulyanov VV. Two–Sided Bounds for PDF’s Maximum of a Sum of Weighted Chi-square Variables. In Shiryaev AN, Samouylov KE, Kozyrev DV, editors, Recent Developments in Stochastic Methods and Applications - ICSM-5, Selected Contributions. Springer. 2021. p. 178-189. (Springer Proceedings in Mathematics and Statistics). doi: 10.1007/978-3-030-83266-7_13

Bobkov, Sergey G. ; Naumov, Alexey A. ; Ulyanov, Vladimir V. / Two–Sided Bounds for PDF’s Maximum of a Sum of Weighted Chi-square Variables. Recent Developments in Stochastic Methods and Applications - ICSM-5, Selected Contributions. editor / Albert N. Shiryaev ; Konstantin E. Samouylov ; Dmitry V. Kozyrev. Springer, 2021. pp. 178-189 (Springer Proceedings in Mathematics and Statistics).

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

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