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July 2019 Extremal theory for long range dependent infinitely divisible processes
Gennady Samorodnitsky, Yizao Wang
Ann. Probab. 47(4): 2529-2562 (July 2019). DOI: 10.1214/18-AOP1318

Abstract

We prove limit theorems of an entirely new type for certain long memory regularly varying stationary infinitely divisible 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 $\alpha$-Fréchet 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.

Citation

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Gennady Samorodnitsky. Yizao Wang. "Extremal theory for long range dependent infinitely divisible processes." Ann. Probab. 47 (4) 2529 - 2562, July 2019. https://doi.org/10.1214/18-AOP1318

Information

Received: 1 March 2017; Revised: 1 May 2018; Published: July 2019
First available in Project Euclid: 4 July 2019

zbMATH: 07114723
MathSciNet: MR3980927
Digital Object Identifier: 10.1214/18-AOP1318

Subjects:
Primary: 60F17, 60G70
Secondary: 60G57

Rights: Copyright © 2019 Institute of Mathematical Statistics

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Vol.47 • No. 4 • July 2019
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