Improvement of characterization inequality for norms of  φ-sub-Gaussian random variable

Dmytro Tykhonenko; Rostyslav Yamnenko; Katalin Kuchinka

doi:10.17721/1812-5409.2025/2.10

Authors

Dmytro Tykhonenko Taras Shevchenko National University of Kyiv
Rostyslav Yamnenko Taras Shevchenko National University of Kyiv https://orcid.org/0000-0002-9612-7959
Katalin Kuchinka Ferenc Rakoczi II Transcarpathian Hungarian University, Berehove, Ukraine; University Tokaj, Sárospatak, Hungary

DOI:

https://doi.org/10.17721/1812-5409.2025/2.10

Keywords:

Stirling's formula, remainder, characterization inequality, norm, φ-sub-Gaussian random variable

Abstract

The paper is focused on refining the characterization inequality for the norm of φ-sub-Gaussian random variables. The core of this improvement lies in a detailed analysis of the remainder terms in Stirling's formula, which provides an approximation for n!. This study conducts a comparative analysis of several pairs of known remainder terms, α_n and β_n, to identify the pair that minimizes the approximation error. By calculating the absolute difference for these pairs, we demonstrate that the tightest bounds are achieved with a specific selection of higher-order terms. This optimized approximation for n! is then applied to sharpen the constant in a characterization inequality for comparing different norms of a φ-sub-Gaussian random variable and also in the theorem on φ-sub-Gaussianity of a random variable.

Pages of the article in the issue: 82 - 86

Language of the article: English

Author Biographies

Dmytro Tykhonenko, Taras Shevchenko National University of Kyiv

PhD Student,

Department of Probability Theory, Statistics and Actuarial Mathematics
Rostyslav Yamnenko, Taras Shevchenko National University of Kyiv

Кафедра теорії ймовірності, статистики і актуарної математики, доцент

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