Positive Semidefinite Supermartingales and Randomized Matrix Concentration Inequalities

We present new concentration inequalities for either martingale dependent or exchangeable random symmetric matrices under a variety of tail conditions, encompassing now-standard Chernoff bounds to self-normalized heavy-tailed settings. These inequalities are often randomized in a way that renders them strictly tighter than existing deterministic results in the literature, are typically expressed in the Loewner order, and are sometimes valid at arbitrary data-dependent stopping times. Along the way, we explore the theory of positive semidefinite supermartingales and maximal inequalities, a natural matrix analog of scalar nonnegative supermartingales that is potentially of independent interest.