engleski

Palm theory for extremes of stationary regularly varying time series and random fields

The tail process $\bY=(Y_{; ; \bi}; ; )_{; ; \bi\in\Z^d}; ; $ of a stationary regularly varying random field $\bX= (X_{; ; \bi}; ; )_{; ; \bi\in\Z^d}; ; $ represents the asymptotic local distribution of $\bX$ as seen from its typical exceedance over a threshold $u$ as $u\to\infty$. Motivated by the standard Palm theory, we show that every tail process satisfies an invariance property called exceedance-stationarity and that this property, together with the spectral decomposition of the tail process, characterizes the class of all tail processes. We then restrict to the case when $Y_{; ; \bi}; ; \to 0$ as $|\bi|\to\infty$ and establish a couple of Palm-like dualities between the tail process and the so-called anchored tail process which, under suitable conditions, represents the asymptotic distribution of a typical cluster of extremes of $\bX$. The main message is that the distribution of the tail process is biased towards clusters with more exceedances. Finally, we use these results to determine the distribution of a typical cluster of extremes for moving average processes with random coefficients and heavy-tailed innovations.

tail process ; regular variation ; Palm theory ; stationarity ; random elds ; time series ; moving averages

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