HIDDEN REGULAR VARIATION FOR POINT PROCESSES AND THE SINGLE/MULTIPLE LARGE POINT HEURISTIC

• 成果类型: Article
• 署名作者: Dombry, Clement; Tillier, Charles; Wintenberger, Olivier
• 署名单位: Universite Marie et Louis Pasteur; Universite Paris Saclay; Universite Paris Cite; Sorbonne Universite
• 刊物名称: ANNALS OF APPLIED PROBABILITY
• ISSN/ISSBN: 1050-5164
• DOI: 10.1214/21-AAP1675
• 发表日期: 2022
• 页码: 191-234
• 关键词: large deviations; extremal behavior

摘要

We consider regular variation for marked point processes with independent heavy-tailed marks and prove a single large point heuristic: the limit measure is concentrated on the cone of point measures with one single point. We then investigate successive hidden regular variation removing the cone of point measures with at most k points, k >= 1, and prove a multiple large point phenomenon: the limit measure is concentrated on the cone of point measures with k + 1 points. We show how these results imply hidden regular variation in Skorokhod space of the associated risk process, in connection with the single/multiple large point heuristic from (Ann. Probab. 47 (2019) 3551-3605). Finally, we provide an application to risk theory in a reinsurance model where the k largest claims are covered and we study the asymptotic behavior of the residual risk.