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Concentration inequalities for matrix martingales in continuous time

Bacry, Emmanuel; Muzy, Jean-François; Gaïffas, Stéphane (2018), Concentration inequalities for matrix martingales in continuous time, Probability Theory and Related Fields, 170, p. 525–553. 10.1007/s00440-017-0786-9

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Type
Article accepté pour publication ou publié
Date
2018
Journal name
Probability Theory and Related Fields
Number
170
Publisher
Springer
Pages
525–553
Publication identifier
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Author(s)
Bacry, Emmanuel cc
CEntre de REcherches en MAthématiques de la DEcision [CEREMADE]
Muzy, Jean-François
Sciences pour l'environnement [SPE]
Gaïffas, Stéphane
Abstract (EN)
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.
Subjects / Keywords
matrix martingales