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Weak convergence of high level crossings and maxima for one or more Gaussian processes

Lindgren, Georg LU ; de Maré, Jacques and Rootzén, Holger LU (1975) In Annals of Probability 3(6). p.961-978
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.
Please use this url to cite or link to this publication:
author
organization
publishing date
type
Contribution to journal
publication status
published
subject
keywords
Stationary Gaussian processes, upcrossings, local maxima, dependent processes, weak convergence
in
Annals of Probability
volume
3
issue
6
pages
961 - 978
publisher
Institute of Mathematical Statistics
ISSN
0091-1798
DOI
10.1214/aop/1176996222
language
English
LU publication?
yes
id
04cc320e-5ee6-4ad0-9191-41ba25faf300 (old id 1273144)
alternative location
http://projecteuclid.org/DPubS/Repository/1.0/Disseminate?view=body&id=pdf_1&handle=euclid.aop/1176996222
date added to LUP
2009-06-03 16:53:14
date last changed
2016-04-16 06:54:47
@misc{04cc320e-5ee6-4ad0-9191-41ba25faf300,
  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.},
  author       = {Lindgren, Georg and de Maré, Jacques and Rootzén, Holger},
  issn         = {0091-1798},
  keyword      = {Stationary Gaussian processes,upcrossings,local maxima,dependent processes,weak convergence},
  language     = {eng},
  number       = {6},
  pages        = {961--978},
  publisher    = {ARRAY(0x9741a58)},
  series       = {Annals of Probability},
  title        = {Weak convergence of high level crossings and maxima for one or more Gaussian processes},
  url          = {http://dx.doi.org/10.1214/aop/1176996222},
  volume       = {3},
  year         = {1975},
}