Phantom distribution functions for some stationary sequences
2015
Doukhan, Paul | Jakubowski, Adam | Lang, Gabriel | Analyse, Géométrie et Modélisation (AGM - UMR 8088) ; Centre National de la Recherche Scientifique (CNRS)-CY Cergy Paris Université (CY) | Nicolaus Copernicus University [Toruń] | Mathématiques et Informatique Appliquées (MIA-Paris) ; Institut National de la Recherche Agronomique (INRA)-AgroParisTech
The notion of a phantom distribution function (phdf) was introduced by O’Brien (Ann. Probab. 15, 281–292 (1987)). We show that the existence of a phdf is a quite common phenomenon for stationary weakly dependent sequences. It is proved that any α-mixing stationary sequence with continuous marginals admits a continuous phdf. Sufficient conditions are given for stationary sequences exhibiting weak dependence, what allows the use of attractive models beyond mixing. The case of discontinuous marginals is also discussed for α-mixing. Special attention is paid to examples of processes which admit a continuous phantom distribution function while their extremal index is zero. We show that Asmussen (Ann. Appl. Probab. 8, 354– 374 1998) and Roberts et al. (Extremes. 9, 213–229 2006) provide natural examples of such processes. We also construct a non-ergodic stationary process of this type.
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