article · The Annals of Probability · 2019
Extremal theory for long range dependent infinitely divisible processes
Gennady Samorodnitsky, Yizao Wang
Cited by 25Open access
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.
Cite this paper
Samorodnitsky, G., & Wang, Y. (2019). Extremal theory for long range dependent infinitely divisible processes. The Annals of Probability, 47(4). https://doi.org/10.1214/18-aop1318
Samorodnitsky, Gennady, and Yizao Wang. “Extremal theory for long range dependent infinitely divisible processes.” The Annals of Probability, vol. 47, no. 4, 2019, https://doi.org/10.1214/18-aop1318.
Samorodnitsky, Gennady, and Yizao Wang. 2019. “Extremal theory for long range dependent infinitely divisible processes.” The Annals of Probability 47 (4). https://doi.org/10.1214/18-aop1318.
Samorodnitsky, G. and Wang, Y. (2019) ‘Extremal theory for long range dependent infinitely divisible processes’, The Annals of Probability, 47(4). Available at: https://doi.org/10.1214/18-aop1318.
@article{samorodnitsky2019extremal,
title = {{Extremal theory for long range dependent infinitely divisible processes}},
author = {Samorodnitsky, Gennady and Wang, Yizao},
journal = {The Annals of Probability},
volume = {47},
number = {4},
year = {2019},
doi = {10.1214/18-aop1318}
}Made from the index record. Check names, capitalisation and page numbers against the paper. Need another style? AI for Research has over 10,000.