On the speed of convergence of discrete Pickands constants to continuous ones
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arXiv
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| Autores principales: | , |
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| Formato: | Preprint |
| Publicado: |
2021
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| _version_ | 1866913696452706304 |
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| author | Bisewski, Krzysztof Jasnovidov, Grigori |
| author_facet | Bisewski, Krzysztof Jasnovidov, Grigori |
| contents | In this manuscript, we address open questions raised by Dieker \& Yakir (2014), who proposed a novel method of estimation of (discrete) Pickands constants $\mathcal{H}^δ_α$ using a family of estimators $ξ^δ_α(T), T>0$, where $α\in(0,2]$ is the Hurst parameter, and $δ\geq0$ is the step-size of the regular discretization grid. We derive an upper bound for the discretization error $\mathcal{H}_α^0 - \mathcal{H}_α^δ$, whose rate of convergence agrees with Conjecture 1 of Dieker & Yakir (2014) in case $α\in(0,1]$ and agrees up to logarithmic terms for $α\in(1,2)$. Moreover, we show that all moments of $ξ_α^δ(T)$ are uniformly bounded and the bias of the estimator decays no slower than $\exp\{-\mathcal CT^α\}$, as $T$ becomes large. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2108_00756 |
| institution | arXiv |
| publishDate | 2021 |
| record_format | arxiv |
| spellingShingle | On the speed of convergence of discrete Pickands constants to continuous ones Bisewski, Krzysztof Jasnovidov, Grigori Probability 60G15, 60G70, 65C05 In this manuscript, we address open questions raised by Dieker \& Yakir (2014), who proposed a novel method of estimation of (discrete) Pickands constants $\mathcal{H}^δ_α$ using a family of estimators $ξ^δ_α(T), T>0$, where $α\in(0,2]$ is the Hurst parameter, and $δ\geq0$ is the step-size of the regular discretization grid. We derive an upper bound for the discretization error $\mathcal{H}_α^0 - \mathcal{H}_α^δ$, whose rate of convergence agrees with Conjecture 1 of Dieker & Yakir (2014) in case $α\in(0,1]$ and agrees up to logarithmic terms for $α\in(1,2)$. Moreover, we show that all moments of $ξ_α^δ(T)$ are uniformly bounded and the bias of the estimator decays no slower than $\exp\{-\mathcal CT^α\}$, as $T$ becomes large. |
| title | On the speed of convergence of discrete Pickands constants to continuous ones |
| topic | Probability 60G15, 60G70, 65C05 |
| url | https://arxiv.org/abs/2108.00756 |