Concentration inequalities for matrix martingales in continuous time

• 成果类型: Article
• 署名作者: Bacry, Emmanuel; Gaiffas, Stephane; Muzy, Jean-Francois
• 署名单位: Institut Polytechnique de Paris; Ecole Polytechnique; ENSTA Paris; Centre National de la Recherche Scientifique (CNRS); CNRS - National Institute for Mathematical Sciences (INSMI); Institut Polytechnique de Paris; Ecole Polytechnique; Centre National de la Recherche Scientifique (CNRS); CNRS - Institute for Engineering & Systems Sciences (INSIS)
• 刊物名称: PROBABILITY THEORY AND RELATED FIELDS
• ISSN/ISSBN: 0178-8051
• DOI: 10.1007/s00440-017-0786-9
• 发表日期: 2018
• 页码: 525-553
• 关键词: exponential inequalities; NORM

摘要

This paper gives new concentration inequalities for the spectral norm of a wide class of matrix martingales in continuous time. These results extend previously established Freedman and Bernstein inequalities for series of random matrices to the class of continuous time processes. Our analysis relies on a new supermartingale property of the trace exponential proved within the framework of stochastic calculus. We provide also several examples that illustrate the fact that our results allow us to recover easily several formerly obtained sharp bounds for discrete time matrix martingales.