EXTREMAL THEORY FOR LONG RANGE DEPENDENT INFINITELY DIVISIBLE PROCESSES

• 成果类型: Article
• 署名作者: Samorodnitsky, Gennady; Wang, Yizao
• 署名单位: Cornell University; University System of Ohio; University of Cincinnati
• 刊物名称: ANNALS OF PROBABILITY
• ISSN/ISSBN: 0091-1798
• DOI: 10.1214/18-AOP1318
• 发表日期: 2019
• 页码: 2529-2562
• 关键词: alpha-stable processes; central-limit-theorem; maximum term; stationary; intersections; CONVERGENCE; memory

摘要

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-Frechet 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 is an element of (0, infinity) and beta is an element of (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 is an element of (0, 2) and beta is an element of (0, 1/2) have been considered.