G. Giorgobiani, V. Kvaratskhelia, M. Menteshashvili
{"title":"Unconditional Convergence of Sub-Gaussian Random Series","authors":"G. Giorgobiani, V. Kvaratskhelia, M. Menteshashvili","doi":"10.1134/s1054661824010061","DOIUrl":null,"url":null,"abstract":"<h3 data-test=\"abstract-sub-heading\">Abstract</h3><p>In this paper we explore the basic properties of sub-Gaussian random variables and random elements. We also present various notions of subgaussianity (weak, <span>\\({\\mathbf{T}}\\)</span>- and <span>\\({\\mathbf{F}}\\)</span>-subgaussianity) of random elements with values in general Banach spaces. It is shown that the covariance operator of <span>\\({\\mathbf{T}}\\)</span>-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.</p>","PeriodicalId":35400,"journal":{"name":"PATTERN RECOGNITION AND IMAGE ANALYSIS","volume":null,"pages":null},"PeriodicalIF":0.7000,"publicationDate":"2024-04-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"PATTERN RECOGNITION AND IMAGE ANALYSIS","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/10.1134/s1054661824010061","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q4","JCRName":"COMPUTER SCIENCE, INTERDISCIPLINARY APPLICATIONS","Score":null,"Total":0}
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.
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.
京公网安备 11010802042870号
京ICP备2023020795号-1
分享
求助内容：
应助结果提醒方式：
扫码关注我们
