High level sojourns for strongly dependent Gaussian processes

Abstract

Let X(t), t>O, be a real, stationary Gaussian stochastic process with mean 0, variance 1, and continuous covariance function r(t). For t > 0 and u > 0, let Lt(u ) be the time spent by X(s), O<s<t, above the level u. (Since r(t) is continuous, X(t) has a measurable version so that Lt(u ) is well defined for this version.) Let u = u(t) increase to oo with t. If u(t) increases at a sufficiently slow rate, and if r(t) is nonnegative for all large t and tends to 0 at a sufficiently slow rate, then Lt(u(t)) has, upon appropriate normalization, a limiting normal distribution. Let Mt(u ) be the time spent by IX(s)], 0 < s < t , above u. Under similar conditions on u(t) but more special conditions on r(t) for t ~ 0% Mt(u(t)) has, upon appropriate normalization, a limiting distribution which is not Gaussian. The latter distribution was discovered by Rosenblatt (1961), and applied by Taqqu (1975). These results belong to the domain of limit theorems for sojourns of stationary Gaussian processes above high levels and over long time intervals. The present state of knowledge may be quickly summarized as follows: If the covariance function r(t) tends to 0 sufficiently rapidly for t--+ 0% and if u(t) is defined as (21ogt) 1/2 plus a term of lower order, then Lt(u(t)) has, after division by its expected value, a limiting distribution which is compound Poisson. This has been established under increasingly general conditions by Volkonskii and Rozanov (1961), Cramer and Leadbetter (1967), and Berman (1971) and (1979). The current paper has an entirely different set of hypotheses and conclusions. We assume, as before, that r(t)-+0 for t -+co; however, we require that this convergence take place at a sufficiently slow rate. Therefore, the parts of the process over subsets of the time parameter set which are widely separated in time are not nearly independent, so that a limiting distribution cannot be obtained by the usual imitation of the case of independence. Another major difference is that the function u(t) is assumed to increase more slowly than in the