Unconditional Convergence of Sub-Gaussian Random Series

IF 0.7 Q4 COMPUTER SCIENCE, INTERDISCIPLINARY APPLICATIONS
G. Giorgobiani, V. Kvaratskhelia, M. Menteshashvili
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引用次数: 0

Abstract

In this paper we explore the basic properties of sub-Gaussian random variables and random elements. We also present various notions of subgaussianity (weak, \({\mathbf{T}}\)- and \({\mathbf{F}}\)-subgaussianity) of random elements with values in general Banach spaces. It is shown that the covariance operator of \({\mathbf{T}}\)-subgaussian random element is Gaussian and some consequences of this result in spaces possessing certain geometric properties are noted. Moreover, the almost sure (a.s.) unconditional convergence of random series are considered and a sufficient condition of a.s. unconditional convergence of a random series of a special type with values in a Banach space with some geometric properties is proved. By the a.s. unconditional convergence of random series we understand the convergence of all rearrangements of the series on the same set of probability 1. With some effort, we prove one of the main results of the paper, which gives us a necessary condition for the a.s. unconditional convergence of random series of a special type in a general Banach space. For the proof, a lemma is used that establishes a connection between the moments of a random variable and which may be of independent interest.

亚高斯随机序列的无条件收敛性
摘要 本文探讨了亚高斯随机变量和随机元素的基本性质。我们还提出了在一般巴拿赫空间中取值的随机元素的亚高斯性(弱、\({\mathbf{T}}\)-和\({\mathbf{F}}\)-亚高斯性)的各种概念。研究表明,\({\mathbf{T}}\)-次高斯随机元素的协方差算子是高斯的,并指出了这一结果在具有某些几何性质的空间中的一些后果。此外,还考虑了随机数列的几乎确定(a.s. )无条件收敛,并证明了在具有某些几何性质的巴拿赫空间中取值的特殊类型随机数列的a.s. 无条件收敛的充分条件。通过随机数列的 a.s.无条件收敛,我们理解了在概率为 1 的同一集合上数列的所有重排的收敛。经过一番努力,我们证明了本文的主要结果之一,它给出了在一般巴拿赫空间中特殊类型随机数列无条件收敛的必要条件。为了证明这一点,我们使用了一个在随机变量的矩之间建立联系的 Lemma,它可能会引起我们的兴趣。
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来源期刊
PATTERN RECOGNITION AND IMAGE ANALYSIS
PATTERN RECOGNITION AND IMAGE ANALYSIS Computer Science-Computer Graphics and Computer-Aided Design
CiteScore
1.80
自引率
20.00%
发文量
80
期刊介绍: The purpose of the journal is to publish high-quality peer-reviewed scientific and technical materials that present the results of fundamental and applied scientific research in the field of image processing, recognition, analysis and understanding, pattern recognition, artificial intelligence, and related fields of theoretical and applied computer science and applied mathematics. The policy of the journal provides for the rapid publication of original scientific articles, analytical reviews, articles of the world''s leading scientists and specialists on the subject of the journal solicited by the editorial board, special thematic issues, proceedings of the world''s leading scientific conferences and seminars, as well as short reports containing new results of fundamental and applied research in the field of mathematical theory and methodology of image analysis, mathematical theory and methodology of image recognition, and mathematical foundations and methodology of artificial intelligence. The journal also publishes articles on the use of the apparatus and methods of the mathematical theory of image analysis and the mathematical theory of image recognition for the development of new information technologies and their supporting software and algorithmic complexes and systems for solving complex and particularly important applied problems. The main scientific areas are the mathematical theory of image analysis and the mathematical theory of pattern recognition. The journal also embraces the problems of analyzing and evaluating poorly formalized, poorly structured, incomplete, contradictory and noisy information, including artificial intelligence, bioinformatics, medical informatics, data mining, big data analysis, machine vision, data representation and modeling, data and knowledge extraction from images, machine learning, forecasting, machine graphics, databases, knowledge bases, medical and technical diagnostics, neural networks, specialized software, specialized computational architectures for information analysis and evaluation, linguistic, psychological, psychophysical, and physiological aspects of image analysis and pattern recognition, applied problems, and related problems. Articles can be submitted either in English or Russian. The English language is preferable. Pattern Recognition and Image Analysis is a hybrid journal that publishes mostly subscription articles that are free of charge for the authors, but also accepts Open Access articles with article processing charges. The journal is one of the top 10 global periodicals on image analysis and pattern recognition and is the only publication on this topic in the Russian Federation, Central and Eastern Europe.
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