The excursion random measure $\zeta$ of a stationary process is
defined on sets $E \subset (-\infty, \infty) \times (0, \infty)$, as the time
which the process (suitably normalized) spends in the set E . Particular cases
thus include a multitude of features (including sojourn times) related to high
levels. It is therefore not surprising that a single limit theorem for $\zeta$
at high levels contains a wide variety of useful extremal and high level
exceedance results for the stationary process itself.
¶ The theory given for the excursion random measure demonstrates,
under very general conditions, its asymptotic infinite divisibility with
certain stability and independence of increments properties leading to its
asymptotic distribution (Theorem 4.1). The results are illustrated by a number
of examples including stable and Gaussian processes.