dc.contributor.author Samorodnitskty, Gennady dc.contributor.author Wang, Yizao dc.date.accessioned 2017-03-22T18:15:45Z dc.date.available 2017-03-22T18:15:45Z dc.date.issued 2017-03 dc.identifier.uri https://hdl.handle.net/1813/47543 dc.description.abstract We prove limit theorems of an entirely new type for certain long memory en_US regularly varying stationary \id\ random processes. These theorems involve multiple phase transitions governed by how long the memory is. Apart from one regime, our results exhibit limits that are not among the classical extreme value distributions. Restricted to the one-dimensional case, the distributions we obtain interpolate, in the appropriate parameter range, the $\alpha$-Fr\'echet distribution and the skewed $\alpha$-stable distribution. In general, the limit is a new family of stationary and self-similar random sup-measures with parameters $\alpha\in(0,\infty)$ and $\beta\in(0,1)$, with representations based on intersections of independent $\beta$-stable regenerative sets. The tail of the limit random sup-measure on each interval with finite positive length is regularly varying with index $-\alpha$. The intriguing structure of these random sup-measures is due to intersections of independent $\beta$-stable regenerative sets and the fact that the number of such sets intersecting simultaneously increases to infinity as $\beta$ increases to one. The results in this paper extend substantially previous investigations where only $\alpha\in(0,2)$ and $\beta\in(0,1/2)$ have been considered. dc.description.sponsorship Samorodnitsky's research was partially supported by the NSF grant en_US DMS-1506783 and the ARO grant W911NF-12-10385 at Cornell University. Wang's research was partially supported by the NSA grants H98230-14-1-0318 and H98230-16-1-0322, and the ARO grant W911NF-17-1-0006 at University of Cincinnati. dc.language.iso en_US en_US dc.subject Extreme value theory en_US dc.subject random sup-measure en_US dc.subject random upper-semi-continuous function en_US dc.subject stationary infinitely divisible process en_US dc.subject long range dependence en_US dc.title Extremal theory for long range dependent infinitely divisible processes en_US dc.type article en_US
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