Asymptotic Expansions of Crossing Rates of Stationary Random Processes

Oskar Hagberg

Centre for Mathematical Sciences
Mathematical Statistics
Lund Institute of Technology

ISBN 91-628-6384-3

The crossing rate of a stationary random process is a valuble tool when studying crest hight distributions and maxima of sea level elevation. This thesis considers two approximation techinques for cases when the crossing rate cannot be exactly computed. Both techniques use an asymptotic expansion, and the first and second order terms in these expansions are given explicitly.
Paper A describes how the rate of crossings is used to study crest hight distributions and maxima of sea level elevation and serves as a motivation for the subsequent three papers. Paper B and C both study an approximation technique proposed by Breitung (1988); Paper B considers the special case of the quadratic form of a Gaussian random process, while Paper C considers the general case that Breitung studied. Papers D treats the so- called Saddle point approximation of the crossing rate, allready studied informally by Butler et al (2003).