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

Matrix concentration inequalities control the deviation of a random matrix around its mean. Until now, results in literature consider the case of sums of independent random matrices, or matrix martingales in discrete time. A first matrix version of the Chernoff bound is given in [1] and was adapted to yield matrix analogues of standard scalar concentration inequalities in [7, 25, 26, 23]. Later, these results were improved in [34, 33], by the use of a theorem due to Lieb [18], about the concavity of a trace exponential function, which is a deep result closely related to the joint convexity of quantum entropy in physics. See also [20] for a family of sharper and more general results based on the Stein's method. These works contain extensions to random matrices of classical concentration inequalities for sums of independent scalar random variables, such as the Bernstein inequality for sub-exponential random variables, Hoeffding inequality for sub-Gaussian random variables or Freedman inequality for martingales, see e.g.