no. 8
Probabilités
The distribution of the maximum of an ARMA(1, 1) process
Withers, Christopher S.1 ; Nadarajah, Saralees2
1 Callaghan Innovation, Lower Hutt, New Zealand
2 University of Manchester, Manchester M13 9PL, UK
Comptes Rendus. Mathématique, Tome 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
Withers, Christopher S. 1 ; Nadarajah, Saralees 2

1 Callaghan Innovation, Lower Hutt, New Zealand
2 University of Manchester, Manchester M13 9PL, UK 
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Withers, Christopher S.; Nadarajah, Saralees. The distribution of the maximum of an ARMA(1, 1) process. Comptes Rendus. Mathématique, Tome 358 (2020) no. 8, pp. 909-916. doi : 10.5802/crmath.111. http://www.numdam.org/articles/10.5802/crmath.111/

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