The distribution of the maximum of an ARMA(1, 1) process

Christopher S. Withers; Saralees Nadarajah

doi:10.5802/crmath.111

Probabilités

The distribution of the maximum of an ARMA(1, 1) process

Christopher S. Withers ¹ ; Saralees Nadarajah ²

¹ Callaghan Innovation, Lower Hutt, New Zealand
² University of Manchester, Manchester M13 9PL, UK

Comptes Rendus. Mathématique, Volume 358 (2020) no. 8, pp. 909-916.

Résumé

We give the cumulative distribution function of $M_{n} = max (X_{1}, ..., X_{n})$ , the maximum of a sequence of $n$ observations from an ARMA(1, 1) process. Solutions are first given in terms of repeated integrals and then for the case, where the underlying random variables are absolutely continuous. The distribution of $M_{n}$ is then given as a weighted sum of the $n$ th powers of the eigenvalues of a non-symmetric Fredholm kernel. The weights are given in terms of the left and right eigenfunctions of the kernel.

These results are large deviations expansions for estimates, since the maximum need not be standardized to have a limit. In fact, such a limit need not exist.

Reçu le : 2020-08-16
Accepté le : 2020-09-03
Publié le : 2020-12-03

DOI : 10.5802/crmath.111

Affiliations des auteurs :

Christopher S. Withers ¹ ; Saralees Nadarajah ²

¹ Callaghan Innovation, Lower Hutt, New Zealand
² University of Manchester, Manchester M13 9PL, UK

Licence :

CC-BY 4.0

Droits d'auteur : Les auteurs conservent leurs droits

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     author = {Christopher S. Withers and Saralees Nadarajah},
     title = {The distribution of the maximum of an {ARMA(1,} 1) process},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {909--916},
     publisher = {Acad\'emie des sciences, Paris},
     volume = {358},
     number = {8},
     year = {2020},
     doi = {10.5802/crmath.111},
     language = {en},
}

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Christopher S. Withers; Saralees Nadarajah. The distribution of the maximum of an ARMA(1, 1) process. Comptes Rendus. Mathématique, Volume 358 (2020) no. 8, pp. 909-916. doi : 10.5802/crmath.111. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.5802/crmath.111/

Bibliographie
Cité par

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[4] Christopher S. Withers; Saralees Nadarajah The distribution of the maximum of a first order autoregressive process: The continuous case, Metrika, Volume 74 (2011) no. 2, pp. 247-266 | DOI | MR | Zbl

[5] Christopher S. Withers; Saralees Nadarajah The distribution of the maximum of a first order moving average: The continuous case, Extremes, Volume 17 (2014) no. 1, pp. 1-24 | DOI | MR | Zbl

[6] Christopher S. Withers; Saralees Nadarajah The distribution of the maximum of a second order autoregressive process: The continuous case, 2020 (Technical Report, Department of Mathematics, University of Manchester, UK)

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