The Annals of Probability

Weak Convergence of High Level Crossings and Maxima for One or More Gaussian Processes

Georg Lindgren, Jacques de Mare, and Holger Rootzen

Full-text: Open access


Weak convergence of the multivariate point process of upcrossings of several high levels by a stationary Gaussian process is established. The limit is a certain multivariate Poisson process. This result is then used to determine the joint asymptotic distribution of heights and locations of the highest local maxima over an increasing interval. The results are generalized to upcrossings and local maxima of two dependent Gaussian processes. To prevent nuisance jitter from hiding the overall structure of crossings and maxima the above results are phrased in terms of $\varepsilon$-crossings and $\varepsilon$-maxima, but it is shown that under suitable regularity conditions the results also hold for ordinary upcrossings and maxima.

Article information

Ann. Probab., Volume 3, Number 6 (1975), 961-978.

First available in Project Euclid: 19 April 2007

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier


Primary: 60G15: Gaussian processes
Secondary: 60F05: Central limit and other weak theorems 60G10: Stationary processes 60G17: Sample path properties

Stationary Gaussian processes upcrossings local maxima dependent processes weak convergence


Lindgren, Georg; de Mare, Jacques; Rootzen, Holger. Weak Convergence of High Level Crossings and Maxima for One or More Gaussian Processes. Ann. Probab. 3 (1975), no. 6, 961--978. doi:10.1214/aop/1176996222.

Export citation