Phantom distribution functions for some stationary sequences
Abstract
The notion of a phantom distribution function (phdf) was introduced by O'Brien (1987). We show that the existence of a phdf is a quite common phenomenon for stationary weakly dependent sequences. It is proved that any -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 -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 (1998) and Roberts et al. (2006) provide natural examples of such processes. We also construct a non-ergodic stationary process of this type.
Cite
@article{arxiv.1509.05449,
title = {Phantom distribution functions for some stationary sequences},
author = {Paul Doukhan and Adam Jakubowski and Gabriel Lang},
journal= {arXiv preprint arXiv:1509.05449},
year = {2015}
}