ASYMPTOTIC BEHAVIOR OF THE EXPECTED LENGTH OF EXCURSIONS ABOVE A FIXED LEVEL FOR SOME ELLIPSOIDAL PROCESSES

摘要

The asymptotic order of the expected length of excursions above a fixed level u for a Gaussian process in continuous time is O(u~(-1)) when u is sufficiently large. The expected length of excursions above level u in discrete time can be defined in a similar manner to the case of continuous time. Asymptotic orders in the discrete case are O(u~(-1)) for the Gaussian process and O(u~0) for the process having the Pearson Type Ⅶ distribution. This paper shows that, for the non-Gaussian processes having generalized Laplace and logistic distributions, their asymptotic orders of the expected length of excursions in the discrete case are equal to O(u~(-1/2)).