  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
$\alpha$-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
$\alpha$-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.
