标量遍历扩散的浓度及其统计意义
Concentration of scalar ergodic diffusions and some statistical implications
摘要：
我们推导了经验过程的连续时间模拟和标量遍历扩散过程的相关随机积分的一致浓度不等式。从而，我们奠定了基础，通常需要为一大类扩散过程的估计过程的超范数性质的研究。在经典的i.i.d.在本文中，Talagrand型集中不等式为统计超范数分析提供了一个关键工具。针对扩散框架中的一个平行替代问题，通过鞅逼近和一般链式方法得到的矩界，给出了一个系统的、完备的求解此类一致浓度不等式的方法.所开发的机器是独立的概率兴趣，可以作为一个起点，调查其他过程，如更一般的马尔可夫过程，特别是多变量或离散观察扩散。作为第一个具体的统计应用，我们分析了超范数误差估计的不变密度的遍历扩散通过自然局部时间估计和经典的非参数核密度估计，分别。
英文摘要：
We derive uniform concentration inequalities for continuous-time analogues of empirical processes and related stochastic integrals of scalar ergodic diffusion processes. Thereby, we lay the foundation typically required for the study of sup-norm properties of estimation procedures for a large class of diffusion processes. In the classical i.i.d. context, a key device for the statistical sup-norm analysis is provided by Talagrand-type concentration inequalities. Aiming for a parallel substitute in the diffusion framework, we present a systematic, self-contained approach to such uniform concentration inequalities via martingale approximation and moment bounds obtained by the generic chaining method. The developed machinery is of independent probabilistic interest and can serve as a starting point for investigations of other processes such as more general Markov processes, in particular multivariate or discretely observed diffusions. As a first concrete statistical application, we analyse the sup-norm error of estimating the invariant density of an ergodic diffusion via the natural local time estimator and the classical nonparametric kernel density estimator, respectively.
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