2015/09/17 by Paul Doukhan, Doukhan, Paul, Adam Jakubowski +3
Mathematics · #60F99 #60G10 #60G70 #FOS: Mathematics #Probability (math.PR) #math.PR #msc:60F99 #msc:60G10 #msc:60G70
paper · pdf · doi:10.48550/arxiv.1509.05449
arxiv created 2015/09/17 · arxiv updated 2015/09/21
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 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 (1998) and Roberts et al. (2006) provide natural examples of such processes. We also construct a non-ergodic stationary process of this type.