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Conditions for the convergence in distribution of stationary normal processes

Leadbetter, M. Ross; Lindgren, Georg LU and Rootzén, Holger (1978) In Stochastic Processes and their Applications 8(2). p.131-139
Abstract
The asymptotic distribution of the maximum Mn=max1=<t=<nξt in a stationary normal sequence ξ1,ξ,... depends on the correlation rt between ξ0 and ξt. It is well known that if rt log t -> 0 as t -> ~ or if Σr2t<~, then the limiting distribution is the same as for a sequence of independent normal variables. Here it is shown that this also follows from a weaker condition, which only puts a restriction on the number of t-values for which rt log t islarge. The condition gives some insight into what is essential for this asymptotic behaviour of maxima. Similar results are obtained for a stationary normal process in continuous time.
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subject
keywords
[Mathematical Subject Codes] Primary 60G15, [Mathematical Subject Codes] 60G10, [Mathematical Subject Codes] Stationary normal sequences, [Mathematical Subject Codes] Stationary normal processes, [Mathematical Subject Codes] Limit distribution for maxima
in
Stochastic Processes and their Applications
volume
8
issue
2
pages
131 - 139
publisher
Elsevier
ISSN
1879-209X
language
English
LU publication?
no
id
88e5c7a8-0356-4cbd-9d95-35fa89ac9d80 (old id 1273157)
alternative location
http://ida.lub.lu.se/cgi-bin/elsevier_local?YMT00110-A-03044149-V0008I02-78900029
date added to LUP
2009-06-03 16:29:58
date last changed
2016-06-29 08:59:48
@article{88e5c7a8-0356-4cbd-9d95-35fa89ac9d80,
  abstract     = {The asymptotic distribution of the maximum Mn=max1=&lt;t=&lt;nξt in a stationary normal sequence ξ1,ξ,... depends on the correlation rt between ξ0 and ξt. It is well known that if rt log t -&gt; 0 as t -&gt; ~ or if Σr2t&lt;~, then the limiting distribution is the same as for a sequence of independent normal variables. Here it is shown that this also follows from a weaker condition, which only puts a restriction on the number of t-values for which rt log t islarge. The condition gives some insight into what is essential for this asymptotic behaviour of maxima. Similar results are obtained for a stationary normal process in continuous time.},
  author       = {Leadbetter, M. Ross and Lindgren, Georg and Rootzén, Holger},
  issn         = {1879-209X},
  keyword      = {[Mathematical Subject Codes] Primary 60G15,[Mathematical Subject Codes] 60G10,[Mathematical Subject Codes] Stationary normal sequences,[Mathematical Subject Codes] Stationary normal processes,[Mathematical Subject Codes] Limit distribution for maxima},
  language     = {eng},
  number       = {2},
  pages        = {131--139},
  publisher    = {Elsevier},
  series       = {Stochastic Processes and their Applications},
  title        = {Conditions for the convergence in distribution of stationary normal processes},
  volume       = {8},
  year         = {1978},
}