On Bernstein type inequalities for stochastic integrals of multivariate point processes
On Bernstein type inequalities for stochastic integrals of multivariate point processes
复制标题
多元点过程随机积分的 Bernstein 型不等式
DOI：
10.1016/j.spa.2018.05.014
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发表时间：
2019
影响因子：
1.4
通讯作者：
Su Zhonggen
中科院分区：
文献类型：
--
作者：
Wang Hanchao;Lin Zhengyan;Su Zhonggen
本文首先利用dolsamans - dade指数公式，在一定条件下得到了多元点过程随机积分的Bernstein型集中不等式，然后利用一般链式论证，导出了一个一致的指数不等式。作为直接结果，我们得到了由一类泛函索引的离散时间鞅序列的上界。最后，我们将均匀指数界应用于非参数最大似然估计，并提供了一个以海灵格距离表示的收敛速度，这是van de Geer（1995）早期工作的改进。
In this paper, we first obtain a Bernstein type of concentration inequality for stochastic integrals of multivariate point processes under some conditions through the Doléans-Dade exponential formula, and then derive a uniform exponential inequality using a generic chaining argument. As a direct consequence, we obtain an upper bound for a sequence of discrete time martingales indexed by a class of functionals. Finally, we apply the uniform exponential bound to nonparametric maximum likelihood estimators and provide a rate of convergence in terms of Hellinger distance, which is an improvement of earlier work of van de Geer (1995).