High-level exceedances of non-stationary processes and irregular sets

At first, we show the convergence to a compound Poisson process of the high-level exceedances point process N-n(B) = Sigma(m/n is an element of B) 1({xi m>un}) 1(A) (m), where xi is a stationary and weakly dependent process, u(n) grows to infinity with n in a suitable way and A subset of N satisfies certain geometrical condition, that includes as particular examples, sets where the size of the border is negligible, periodic sets and level sets (i.e., random sets of the form {m is an element of N : xi(m) (omega) is an element of B}, with B a Borel set) of a process xi that satisfies some ergodic properties. At second, we apply this result to non-stationary processes of the form X-m = phi(xi(m), Y-m), where xi and Y are independent, xi is stationary and weakly dependent, Y is non-stationary and satisfies certain ergodic conditions, and phi is a suitable function: we obtain that the high-level exceedances process of X converges to a compound Poisson process. (C) Academie des Sciences/Elsevier, Paris.