Concentration for matrix martingales in continuous time and microscopic activity of social networks
This paper gives new concentration inequalities for the spectral norm of matrix martingales in continuous time. Both cases of purely discountinuous and continuous martingales are considered. The analysis is based on a new supermartingale property of the trace exponential, based on tools from stochastic calculus. Matrix martingales in continuous time are probabilistic objects that naturally appear for statistical learning of time-dependent systems. We focus here on the the microscopic study of (social) networks, based on self-exciting counting processes, such as the Hawkes process, together with a low-rank prior assumption of the self-exciting component. A consequence of these new concentration inequalities is a push forward of the theoretical analysis of such models.