{"title":"Necessary and sufficient conditions for limit theorems for quadratic variations of Gaussian sequences","authors":"L. Viitasaari","doi":"10.1214/15-PS267","DOIUrl":null,"url":null,"abstract":"The quadratic variation of Gaussian processes plays an important role\r\nin both stochastic analysis and in applications such as estimation of\r\nmodel parameters, and for this reason the topic has been extensively\r\nstudied in the literature. In this article we study the convergence of\r\nquadratic sums of general Gaussian sequences. We provide necessary and\r\nsufficient conditions for different types of convergence including\r\nconvergence in probability, almost sure convergence, $L^{p}$-convergence\r\nas well as weak convergence. We use a practical and simple approach\r\nwhich simplifies the existing methodology considerably. As an\r\napplication, we show how convergence of the quadratic variation of a\r\ngiven process can be obtained by an appropriate choice of the underlying\r\nsequence.\r\n<script type=\"text/javascript\"\r\nsrc=\"//cdn.mathjax.org/mathjax/latest/MathJax.js?config=TeX-AMS-MML_HTMLorMML\">","PeriodicalId":46216,"journal":{"name":"Probability Surveys","volume":null,"pages":null},"PeriodicalIF":1.3000,"publicationDate":"2019-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"12","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Probability Surveys","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/10.1214/15-PS267","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q2","JCRName":"STATISTICS & PROBABILITY","Score":null,"Total":0}
引用次数: 12
Abstract
The quadratic variation of Gaussian processes plays an important role
in both stochastic analysis and in applications such as estimation of
model parameters, and for this reason the topic has been extensively
studied in the literature. In this article we study the convergence of
quadratic sums of general Gaussian sequences. We provide necessary and
sufficient conditions for different types of convergence including
convergence in probability, almost sure convergence, $L^{p}$-convergence
as well as weak convergence. We use a practical and simple approach
which simplifies the existing methodology considerably. As an
application, we show how convergence of the quadratic variation of a
given process can be obtained by an appropriate choice of the underlying
sequence.
京公网安备 11010802042870号
京ICP备2023020795号-1
