纯断鞅的指数型集中不等式献给林正炎教授80华诞

Exponential concentration inequalities for purely discontinuous martingales

本文研究与纯断局部鞅相关的指数型集中不等式;通过构造一个新颖的指数鞅并利用停止定理,建立关于纯断局部鞅的一般化集中不等式.作为推论,本文在不同条件下得到了一系列指数型不等式,包括纯断局部鞅的Bernstein型不等式和de la Pena不等式等.此外,本文考虑连续时间矩阵值局部鞅,通过构造迹指数上鞅,建立矩阵值纯断局部鞅算子范数的集中不等式,改进了Bacry等(2018)的结果.

A novel exponential concentration inequality is obtained for purely discontinuous local martingales.The proof is largely based on a new exponential martingale and the optional time rule.As direct applications,we can derive the classical Bernstein type inequality,de la Pena’s inequality and the exponential concentration inequality for purely discontinuous local martingales under the exponential moment or bounded jump assumption.Besides,we consider the continuous time matrix-valued local martingales and obtain a refined concentration inequality for norms of matrix operators through a new exponential supermartingale for traces,which is an improvement of Bacry et al.(2018).