Wave-Length and Amplitude for a Stationary Process after a High Maximum: Decreasing Covariance Function

摘要

The paper is a direct continuation of the paper Wave-length and amplitude for a stationary process after a high maximum, this series No. 742. It deals with the limiting properties as u approaches infinity of wave-length tau sub u and amplitude delta sub u after a local maximum with height u in a stationary normal process. It is assumed that the covariance function r(t) is strictly decreasing for t > 0. The limiting distribution of tau sub u is derived and expressed by means of a certain time transformation and it includes a slight generalization of the Poisson limit theorem for crossings of a very high level. Especially it is shown that tau sub u approaches infinity and that delta sub u/u approaches 1 as u approaches infinity. (Author)