RESEARCH ARTICLE

Maxima and sum for discrete and continuous time Gaussian processes

  • Yang CHEN 1 ,
  • Zhongquan TAN , 2
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  • 1. School of Mathematics and Physics, Suzhou University of Science and Technology,Suzhou 215009, China
  • 2. College of Mathematics, Physics and Information Engineering, Jiaxing University,Jiaxing 314001, China

Received date: 26 Oct 2014

Accepted date: 14 May 2015

Published date: 02 Dec 2015

Copyright

2014 Higher Education Press and Springer-Verlag Berlin Heidelberg

Abstract

We study the asymptotic relation among the maximum of continuous weakly and strongly dependent stationary Gaussian process, the maximum of this process sampled at discrete time points, and the partial sum of this process. It is shown that these two extreme values and the sum are asymptotically independent if the grid of the discrete time points is sufficiently sparse and the Gaussian process is weakly dependent, and asymptotically dependent if the grid points are Pickands grids or dense grids.

Cite this article

Yang CHEN , Zhongquan TAN . Maxima and sum for discrete and continuous time Gaussian processes[J]. Frontiers of Mathematics in China, 2016 , 11(1) : 27 -46 . DOI: 10.1007/s11464-015-0491-x

1
Hashorva E, Tan Z Q. Piterbarg’s max-discretisation theorem for stationary vector Gaussian processes observed on different grids. Statistics, 2015, 49: 338–360

DOI

2
Ho H C, Hsing T. On the asymptotic joint distribution of the sum and maximum of stationary normal random variables. J Appl Probab, 1996, 33: 138–145

DOI

3
Ho H C, McCormick W P. Asymptotic distribution of sum and maximum for Gaussian processes. J Appl Probab, 1999, 36: 1031–1044

DOI

4
Hu A P, Peng Z X, Qi Y C. Joint behavior of point process of exceedances and partial sum from a Gaussian sequence. Metrika, 2009, 70: 279–295

DOI

5
Hüsler J. Dependence between extreme values of discrete and continuous time locally stationary Gaussian processes. Extremes, 2004, 7: 179–190

DOI

6
Hüsler J, Piterbarg V. Limit theorem for maximum of the storage process with fractional Brownian motion as input. Stochastic Process Appl, 2004, 114: 231–250

DOI

7
Leadbetter M R, Lindgren G, Rootz′en H. Extremes and Related Properties of Random Sequences and Processes. Series in Statistics. New York: Springer, 1983

DOI

8
McCormick W P, Qi Y. Asymptotic distribution for the sum and maximum of Gaussian processes. J Appl Probab, 2000, 37: 958–971

DOI

9
Mittal Y, Ylvisaker D. Limit distribution for the maximum of stationary Gaussian processes. Stochastic Process Appl, 1975, 3: 1–18

DOI

10
Peng Z X, Nadarajah S. On the joint limiting distribution of sums and maxima of stationary normal sequence. Theory Probab Appl, 2003, 47: 706–708

DOI

11
Peng Z X, Tong J J, Weng Z C. Joint limit distributions of exceedances point processes and partial sums of Gaussian vector sequence. Acta Math Sin (Engl Ser), 2012, 28: 1647–1662

DOI

12
Pickands J III. Asymptotic properties of the maximum in a stationary Gaussian process. Trans Amer Math Soc, 1969, 145: 75–86

13
Piterbarg V I. Asymptotic Methods in the Theory of Gaussian Processes and Fields. Providence: Amer Math Soc, 1996

14
Piterbarg V I. Discrete and continuous time extremes of Gaussian processes. Extremes, 2004, 7: 161–177

DOI

15
Tan Z Q. The asymptotic relation between the maxima and sums of discrete and continuous time strongly dependent Gaussian processes. Acta Math Appl Sin, 2015,38: 27–36 (in Chinese)

16
Tan Z Q, Hashorva E. On Piterbarg’s max-discretisation theorem for multivariate stationary Gaussian processes. J Math Anal Appl, 2014, 409: 299–314

DOI

17
Tan Z Q, Hashorva E, Peng Z Q. Asymptotics of maxima of strongly dependent Gaussian processes. J Appl Probab, 2012, 49: 1106–1118

DOI

18
Tan Z Q, Tang L J. The dependence of extremes values of discrete and continuous time strongly dependent Gaussian processes. Stochastics, 2014, 86: 60–69

DOI

19
Tan Z Q, Wang K Y. On Piterbarg’s max-discretisation theorem for homogeneous Gaussian random fields. J Math Anal Appl, 2015, 429: 969–994

DOI

20
Tan Z Q, Wang Y B. Extremes values of discrete and continuous time strongly dependent Gaussian processes, Comm Statist Theory Methods, 2013, 42: 2451–2463

DOI

21
Turkman K F. Discrete and continuous time extremes of stationary processes. In: Rao T S, Rao S S, Rao C R, eds. Handbook of Statistics, Vol 30. Time Series Methods and Applications. Amsterdam: Elsevier, 2012, 565–580

DOI

22
Turkman K F, Turkman M A A, Pereira J M. Asymptotic models and inference for extremes of spatio-temporal data. Extremes, 2010, 13: 375–397

DOI

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