论离散皮康兹常数向连续常数的收敛速度

IF 0.7 4区 数学 Q3 STATISTICS & PROBABILITY
Krzysztof Bisewski, Grigori Jasnovidov
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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>. 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引用次数: 0

摘要

在本手稿中,我们将讨论 Dieker 和 Yakir(2014 年)提出的开放性问题,他们提出了一种新方法,即使用估计器系列 $\xi^\delta_\alpha(T)$ , $T>0$ 来估计(离散)皮克兰常数 $\mathcal{H}^\delta_\alpha$ ,其中 $\alpha\in(0,2]$ 是赫斯特参数,$\delta\geq0$ 是常规离散网格的步长。我们推导出离散化误差 $\mathcal{H}_\alpha^0 - \mathcal{H}_\alpha^\delta$的上界,其收敛率在$\alpha/in(0,1]$情况下与Dieker和Yakir(2014)的猜想1一致,在$\alpha/in(1,2)$情况下与对数项一致。此外,我们还证明了$\xi_\alpha^\delta(T)$的所有矩都是均匀有界的,并且当T变大时,估计器偏差的衰减速度不会慢于$\exp\{-\mathcal CT^{\alpha}\}$ 。
本文章由计算机程序翻译,如有差异,请以英文原文为准。
On the speed of convergence of discrete Pickands constants to continuous ones
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.
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来源期刊
Journal of Applied Probability
Journal of Applied Probability 数学-统计学与概率论
CiteScore
1.50
自引率
10.00%
发文量
92
审稿时长
6-12 weeks
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.
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