Concentration for matrix martingales in continuous time and microscopic activity of social networks

TL;DR

This paper develops new concentration inequalities for the spectral norm of continuous-time matrix martingales, extending classical results and enabling sharper bounds in stochastic processes and social network analysis.

Contribution

It introduces a novel supermartingale approach for continuous-time matrix martingales, extending Freedman and Bernstein inequalities to this setting.

Findings

New concentration inequalities for spectral norms of matrix martingales in continuous time

Extension of classical inequalities to continuous-time processes

Application to social network activity analysis

Abstract

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.