## The Annals of Probability

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

#### Abstract

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

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

Dates
First available in Project Euclid: 19 April 2007

https://projecteuclid.org/euclid.aop/1176996222

Digital Object Identifier
doi:10.1214/aop/1176996222

Mathematical Reviews number (MathSciNet)
MR385997

Zentralblatt MATH identifier
0328.60021

JSTOR

#### Citation

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. https://projecteuclid.org/euclid.aop/1176996222