Phantom distribution functions for some stationary sequences

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.