The Rate of Crossings of a Quadratic Form of an ndimensional Stationary
Gaussian Process
Oskar Hagberg
Centre for Mathematical Sciences
Mathematical Statistics
Lund Institute of Technology,
Lund University,
2004
ISSN 14039338

Abstract:

Consider a quadratic form of a vector valued differentiable stationary Gaussian
process. The crossing intensity of a fixed level depends on the joint correlation
structure of the process and its derivative, but no simple exact form is
known for the general case. We give the first and second terms in an asymptotic
expansion, which is valid as the level tends to infinity, and show how to
find higher order terms.


As a byproduct of the proof we see that the crossing intensity can be written
as an integral which, even if it cannot be solved by a simple formula, is
easily evaluated by Monte Carlo integration. We give a simulation scheme
to describe the steps in this procedure.



Key words:

crossings, quadratic form, asymptotic expansion, Gaussian process
