  We give necessary and sufficient conditions for the existence of a phantom
distribution function for a stationary random field on a regular lattice. We
also introduce a less demanding notion of a directional phantom distribution,
with potentially broader area of applicability. Such approach leads to
sectorial limit properties, a phenomenon well-known in limit theorems for
random fields. An example of a stationary Gaussian random field is provided
showing that the two notions do not coincide. Criteria for the existence of the
corresponding notions of the extremal index and the sectorial extremal index
are also given.
