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\title{Phantom distribution for non-stationary time series as an averaging operation}
\author{Adam Jakubowski\\ Nicolaus Copernicus University, Toruń, Poland\\ adjakubo@mat.umk.pl}
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\begin{abstract}

The notion of a phantom distribution function was introduced by O'Brien (1987). Doukhan et al. \cite{DJL15} showed that the existence of a continuous phantom distribution function is a quite common phenomenon among stationary sequences. 

It should be noted that phantom distribution functions may exist also for non-stationary sequences and that the corresponding theory was developed in  \cite{Jak93} and perfected in \cite{JaTr18}.

In the present lecture we discuss several examples of {\em non-stationary} sequences admitting a phantom distribution functions (including e.g. \cite{Hus86}). 
The conclusion is that if a  phantom distribution function exists in the non-stationary case, it admits an interpretation as an averaging operator.

We discuss also other asymptotic representations for maxima of non-stationary sequences.  

This is {\em a joint work} with Paul Doukhan. 
\end{abstract}


\begin{thebibliography}{2}

\bibitem{DJL15} 
P. Doukhan, A. Jakubowski and G. Lang. Phantom distribution functions for some stationary sequences.
{\em Extremes} {\bf 18 (2015)},  697–-725.
\bibitem{Hus86}
J. H\"{u}sler. Extreme Values of Non-Stationary Random Sequences.
{\em Journal of Applied Probability} {\bf  23 (1986)}, 937--950.
\bibitem{Jak93}
A. Jakubowski. An asymptotic independent representation in limit theorems for maxima of nonstationary
random sequences. {\em The Annals of Probability} {\bf  21 (1993)}, 819–-830.
\bibitem{JaTr18}
A. Jakubowski and P. Truszczy\'nski. Quenched phantom distribution functions for Markov chains. Statistics and Probability Letters, {\bf 137 (2018)}, 79--83. 
\end{thebibliography}

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