Two-sided bounds for PDF's maximum of a sum of weighted chi-square variables

S. Bobkov; A. Naumov; V. V. Ulyanov

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Two-sided bounds for PDF's maximum of a sum of weighted chi-square variables

P. 39–42.

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.

Language: English

Keywords: weighted sums Two--sided bounds chi- square variable

Publication based on the results of:

Uncertainty quantification in high-dimensional models (2020)

In book

Сборник материалов V-й Международной конференции по стохастическим методам: The 5th International Conference on Stochastic Methods (ICSM5). 23-27 November 2020, Russia, Moscow.

M.: RUDN, 2020.

A Multivariate CLT for Weighted Sums with Rate of Convergence of Order O(1/n)

Ulyanov V. V., Ayvazyan S., Springer Proceedings in Mathematics & Statistics 2023 No. 425 P. 225–257

The “typical” asymptotic behavior of the weighted sums of independent random vectors in k-dimensional space is considered. It is shown that in this case the rate of convergence in the multivariate central limit theorem is of order O(1/n). This extends the one-dimensional Klartag and Sodin (2011) result ...

Added: December 1, 2023

Inference via randomized test statistics

Nikita Puchkin, Vladimir Ulyanov, Annales de l'institut Henri Poincare (B) Probability and Statistics 2023 Vol. 59 No. 3 P. 1508–1529

We show that external randomization may enforce the convergence of test statistics to their limiting distributions in particular cases. This results in a sharper inference. Our approach is based on a central limit theorem for weighted sums. We apply our method to a family of rank-based test statistics and a family of phi-divergence test statistics ...

Added: September 3, 2023

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

Bobkov S., Naumov A., Ulyanov V. V., , in: Recent Developments in Stochastic Methods and Applications: ICSM-5, Moscow, Russia, November 23–27, 2020, Selected ContributionsVol. 371.: Springer, 2021. P. 178–189.

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, ...

Added: August 5, 2021

A multivariate central limit theorem for weighted sums

Ayvazyan S. A., Ulyanov V. V., , in: Сборник материалов V-й Международной конференции по стохастическим методам: The 5th International Conference on Stochastic Methods (ICSM5). 23-27 November 2020, Russia, Moscow.: M.: RUDN, 2020. P. 21–23.

We consider the typical behavior of the weighted sums of independent identically distributed random vectors in k-dimensional space. It is shown that in this case the rate of convergence in the multivariate central limit theorem is of order O(1/n) up to logarithmic factor. This extends the one-dimensional Klartag and Sodin result. ...

Added: March 23, 2021

KHINCHINE’S THEOREM AND EDGEWORTH APPROXIMATIONS FOR WEIGHTED SUMS

Bobkov S., Annals of Statistics 2019 Vol. 47 No. 3 P. 1616–1633

Let F_n denote the distribution function of the normalized sum of ni.i.d. random variables. In this paper, polynomial rates of approx- imation of Fn by the corrected normal laws are considered in the model where the underlying distribution has a convolution structure. As a basic tool, the convergence part of Khinchine’s theorem in met- ric theory of Diophantine approximations is ...

Added: October 19, 2018