  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.
