Concentration for matrix martingales in continuous time and microscopic activity of social networks
This provides theoretical tools for analyzing continuous-time random matrix processes, with applications in fields like social network activity modeling, though it appears incremental as an extension of existing discrete-time results.
The paper develops new concentration inequalities for the spectral norm of continuous-time matrix martingales, extending previous results like Freedman and Bernstein inequalities to continuous processes. The analysis uses a novel supermartingale property of the trace exponential in stochastic calculus, enabling recovery of sharp bounds for discrete-time cases.
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.