{"title":"On the speed of convergence of discrete Pickands constants to continuous ones","authors":"Krzysztof Bisewski, Grigori Jasnovidov","doi":"10.1017/jpr.2024.37","DOIUrl":null,"url":null,"abstract":"In this manuscript, we address open questions raised by Dieker and Yakir (2014), who proposed a novel method of estimating (discrete) Pickands constants <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0021900224000378_inline1.png\"/> <jats:tex-math> $\\mathcal{H}^\\delta_\\alpha$ </jats:tex-math> </jats:alternatives> </jats:inline-formula> using a family of estimators <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0021900224000378_inline2.png\"/> <jats:tex-math> $\\xi^\\delta_\\alpha(T)$ </jats:tex-math> </jats:alternatives> </jats:inline-formula>, <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0021900224000378_inline3.png\"/> <jats:tex-math> $T>0$ </jats:tex-math> </jats:alternatives> </jats:inline-formula>, where <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0021900224000378_inline4.png\"/> <jats:tex-math> $\\alpha\\in(0,2]$ </jats:tex-math> </jats:alternatives> </jats:inline-formula> is the Hurst parameter, and <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0021900224000378_inline5.png\"/> <jats:tex-math> $\\delta\\geq0$ </jats:tex-math> </jats:alternatives> </jats:inline-formula> is the step size of the regular discretization grid. We derive an upper bound for the discretization error <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0021900224000378_inline6.png\"/> <jats:tex-math> $\\mathcal{H}_\\alpha^0 - \\mathcal{H}_\\alpha^\\delta$ </jats:tex-math> </jats:alternatives> </jats:inline-formula>, whose rate of convergence agrees with Conjecture 1 of Dieker and Yakir (2014) in the case <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0021900224000378_inline7.png\"/> <jats:tex-math> $\\alpha\\in(0,1]$ </jats:tex-math> </jats:alternatives> </jats:inline-formula> and agrees up to logarithmic terms for <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0021900224000378_inline8.png\"/> <jats:tex-math> $\\alpha\\in(1,2)$ </jats:tex-math> </jats:alternatives> </jats:inline-formula>. Moreover, we show that all moments of <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0021900224000378_inline9.png\"/> <jats:tex-math> $\\xi_\\alpha^\\delta(T)$ </jats:tex-math> </jats:alternatives> </jats:inline-formula> are uniformly bounded and the bias of the estimator decays no slower than <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0021900224000378_inline10.png\"/> <jats:tex-math> $\\exp\\{-\\mathcal CT^{\\alpha}\\}$ </jats:tex-math> </jats:alternatives> </jats:inline-formula>, as <jats:italic>T</jats:italic> becomes large.","PeriodicalId":50256,"journal":{"name":"Journal of Applied Probability","volume":null,"pages":null},"PeriodicalIF":0.7000,"publicationDate":"2024-07-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Journal of Applied Probability","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.1017/jpr.2024.37","RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q3","JCRName":"STATISTICS & PROBABILITY","Score":null,"Total":0}
引用次数: 0
Abstract
In this manuscript, we address open questions raised by Dieker and Yakir (2014), who proposed a novel method of estimating (discrete) Pickands constants $\mathcal{H}^\delta_\alpha$ using a family of estimators $\xi^\delta_\alpha(T)$ , $T>0$ , where $\alpha\in(0,2]$ is the Hurst parameter, and $\delta\geq0$ is the step size of the regular discretization grid. We derive an upper bound for the discretization error $\mathcal{H}_\alpha^0 - \mathcal{H}_\alpha^\delta$ , whose rate of convergence agrees with Conjecture 1 of Dieker and Yakir (2014) in the case $\alpha\in(0,1]$ and agrees up to logarithmic terms for $\alpha\in(1,2)$ . Moreover, we show that all moments of $\xi_\alpha^\delta(T)$ are uniformly bounded and the bias of the estimator decays no slower than $\exp\{-\mathcal CT^{\alpha}\}$ , as T becomes large.
期刊介绍:
Journal of Applied Probability is the oldest journal devoted to the publication of research in the field of applied probability. It is an international journal published by the Applied Probability Trust, and it serves as a companion publication to the Advances in Applied Probability. Its wide audience includes leading researchers across the entire spectrum of applied probability, including biosciences applications, operations research, telecommunications, computer science, engineering, epidemiology, financial mathematics, the physical and social sciences, and any field where stochastic modeling is used.
A submission to Applied Probability represents a submission that may, at the Editor-in-Chief’s discretion, appear in either the Journal of Applied Probability or the Advances in Applied Probability. Typically, shorter papers appear in the Journal, with longer contributions appearing in the Advances.