  We investigate extreme value theory for physical systems with a global
conservation law which describe renewal processes, mass transport models and
long-range interacting spin models. As shown previously, a special feature is
that the distribution of the extreme value exhibits a non-analytical point in
the middle of the support. We expose exact relationships between constrained
extreme value theory and well-known quantities of the underlying stochastic
dynamics, all valid beyond the midpoint in generality, i.e. even far from the
thermodynamic limit. For example for renewal processes, the distribution of the
maximum time between two renewal events is exactly related to the mean number
of these events. In the thermodynamic limit, we show how our theory is suitable
to describe typical and rare events which deviate from classical extreme value
theory. For example for the renewal process, we unravel dual scaling of the
extreme value distribution, pointing out two types of limiting laws: a
normalisable scaling function for the typical statistics and a non-normalised
state describing the rare events.
