离散Pickands常数收敛到连续Pickands常数的速度
On the speed of convergence of discrete Pickands constants to continuous ones
AI总结：
本文解决了Dieker & Yakir（2014）提出的开放问题，推导了离散Pickands常数与连续Pickands常数之间离散化误差的上界，证明其收敛速度与猜想一致，并验证了估计量的矩一致有界性及偏差衰减速度。
摘要：
本文解决了Dieker & Yakir（2014）提出的开放问题，他们提出了一种估计（离散）Pickands常数$\mathcal{H}^δ_α$的新方法，使用估计量族$ξ^δ_α(T), T>0$，其中$α\in(0,2]$是Hurst参数，$δ\geq0$是正则离散化网格的步长。我们推导了离散化误差$\mathcal{H}_α^0 - \mathcal{H}_α^δ$的上界，其收敛速度在$α\in(0,1]$时与Dieker & Yakir（2014）的猜想1一致，在$α\in(1,2)$时与猜想1的对数项一致。此外，我们证明$ξ_α^δ(T)$的所有矩都是一致有界的，且当$T$增大时，估计量的偏差衰减速度不慢于$\exp\{-\mathcal CT^α\}$。
英文摘要：
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.
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