Bounds and asymptotic expansions for the distribution of the Maximum of a smooth stationary Gaussian process
Azaïs, Jean-Marc; Cierco-Ayrolles, Christine; Croquette, Alain; Azaïs Jean-Marc; Laboratoire de Statistique et Probabilités, UMR C55830 du CNRS, Université Paul Sabatier, 118 route de Narbonne, 31062 Toulouse Cedex 4, France.; Cierco-Ayrolles Christine; Institut National de la Recherche Agronomique, Unité de Biométrie et Intelligence Artificielle, BP. 27, Chemin de Borde-Rouge, 31326 Castanet-Tolosan Cedex, France.; Laboratoire de Statistique et Probabilités, UMR C55830 du CNRS, Université Paul Sabatier, 118 route de Narbonne, 31062 Toulouse Cedex 4, France.; Croquette Alain; Laboratoire de Statistique et Probabilités, UMR C55830 du CNRS, Université Paul Sabatier, 118 route de Narbonne, 31062 Toulouse Cedex 4, France.
Журнал:
ESAIM: Probability and Statistics
Дата:
1999
Аннотация:
This paper uses the Rice method [18] to give bounds to the distribution of the maximum of a smooth stationary Gaussian process. We give simpler expressions of the first two terms of the Rice series [3,13] for the distribution of the maximum. Our main contribution is a simpler form of the second factorial moment of the number of upcrossings which is in some sense a generalization of Steinberg et al.'s formula ([7] p. 212). Then, we present a numerical application and asymptotic expansions that give a new interpretation of a result by Piterbarg [15].
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