Comptes Rendus
no. 8
Probabilités
The distribution of the maximum of an ARMA(1, 1) process
Christopher S. Withers1 ; Saralees Nadarajah2
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.

We give the cumulative distribution function of M n =maxX 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 nth 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 :
Accepté le :
Publié le :
DOI : 10.5802/crmath.111
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 
@article{CRMATH_2020__358_8_909_0,
     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},
}
TY  - JOUR
AU  - Christopher S. Withers
AU  - Saralees Nadarajah
TI  - The distribution of the maximum of an ARMA(1, 1) process
JO  - Comptes Rendus. Mathématique
PY  - 2020
SP  - 909
EP  - 916
VL  - 358
IS  - 8
PB  - Académie des sciences, Paris
DO  - 10.5802/crmath.111
LA  - en
ID  - CRMATH_2020__358_8_909_0
ER  - 
%0 Journal Article
%A Christopher S. Withers
%A Saralees Nadarajah
%T The distribution of the maximum of an ARMA(1, 1) process
%J Comptes Rendus. Mathématique
%D 2020
%P 909-916
%V 358
%N 8
%I Académie des sciences, Paris
%R 10.5802/crmath.111
%G en
%F CRMATH_2020__358_8_909_0
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/

[1] Handbook of Mathematical Functions (Milton Abramowitz; Irene A. Stegun, eds.), Applied Mathematics Series, 55, U.S. Department of Commerce, National Bureau of Standards, 1964 | Zbl

[2] Holger Rootzén The rate of convergence of extremes of stationary normal sequences, Adv. Appl. Probab., Volume 15 (1983), pp. 54-80 | DOI | MR | Zbl

[3] Holger Rootzén Extreme value theory for moving average processes, Ann. Probab., Volume 14 (1986), pp. 612-652 | DOI | MR | Zbl

[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)

Cité par Sources :

Commentaires - Politique

Ces articles pourraient vous intéresser

A note on bias reduction

Christopher S. Withers; Saralees Nadarajah

C. R. Math (2020)

Moment inequalities for positive random variables

Christopher S. Withers; Saralees Nadarajah

C. R. Math (2010)