Phantom distribution functions for some stationary sequences

TL;DR

This paper explores the prevalence of phantom distribution functions in stationary weakly dependent sequences, providing conditions for their existence and examining cases with discontinuous marginals and zero extremal index.

Contribution

It demonstrates that many stationary weakly dependent sequences admit phantom distribution functions, extending understanding beyond mixing conditions and including discontinuous marginals.

Findings

Any α-mixing stationary sequence with continuous marginals admits a continuous phdf.

Conditions are provided for sequences with weak dependence to have phantom distribution functions.

Examples of processes with a continuous phdf but zero extremal index are discussed.

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 $\alpha$-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.