  We prove an effective variant of the Kazhdan-Margulis theorem generalized to
stationary actions of semisimple groups over local fields: the probability that
the stabilizer of a random point admits a non-trivial intersection with a small
$r$-neighborhood of the identity is at most $\beta r^\delta$ for some explicit
constants $\beta, \delta > 0$ depending only the group. This is a consequence
of a key convolution inequality. We deduce that vanishing at infinity of
injectivity radius implies finiteness of volume. Further applications are the
compactness of the space of discrete stationary random subgroups and a novel
proof of the fact that all lattices in semisimple groups are weakly cocompact.
