{"title":"Asymptotic Behavior of Point Processes of Exceeding the High Levels of Gaussian Stationary Sequence","authors":"V. I. Piterbarg","doi":"10.3103/s0027132223060050","DOIUrl":null,"url":null,"abstract":"<h3 data-test=\"abstract-sub-heading\">Abstract</h3><p>The paper studies the asymptotic behavior of point processes of exits of a Gaussian stationary sequence beyond a level tending to infinity more slowly than in the Poisson limit theorem for the number of exits. Convergence in variation of such point processes to a marked Poisson process is proved. The results of Yu.V. Prokhorov on the best approximation of the Bernoulli distribution by a mixture of Gaussian and Poisson distributions are applied. A.N. Kolmogorov proposed this problem in the early 1950s.</p>","PeriodicalId":42963,"journal":{"name":"Moscow University Mathematics Bulletin","volume":null,"pages":null},"PeriodicalIF":0.2000,"publicationDate":"2024-03-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Moscow University Mathematics Bulletin","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/10.3103/s0027132223060050","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q4","JCRName":"MATHEMATICS","Score":null,"Total":0}
引用次数: 0
Abstract
The paper studies the asymptotic behavior of point processes of exits of a Gaussian stationary sequence beyond a level tending to infinity more slowly than in the Poisson limit theorem for the number of exits. Convergence in variation of such point processes to a marked Poisson process is proved. The results of Yu.V. Prokhorov on the best approximation of the Bernoulli distribution by a mixture of Gaussian and Poisson distributions are applied. A.N. Kolmogorov proposed this problem in the early 1950s.
期刊介绍：
Moscow University Mathematics Bulletin is the journal of scientific publications reflecting the most important areas of mathematical studies at Lomonosov Moscow State University. The journal covers research in theory of functions, functional analysis, algebra, geometry, topology, ordinary and partial differential equations, probability theory, stochastic processes, mathematical statistics, optimal control, number theory, mathematical logic, theory of algorithms, discrete mathematics and computational mathematics.
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