On the excursion random measure of stationary processes

• 成果类型: Article
• 署名作者: Hsing, TL; Leadbetter, MR
• 署名单位: Texas A&M University System; Texas A&M University College Station; University of North Carolina; University of North Carolina Chapel Hill
• 刊物名称: ANNALS OF PROBABILITY
• ISSN/ISSBN: 0091-1798
• 发表日期: 1998
• 页码: 710-742
• 关键词: sojourns; extremes; limit

摘要

The excursion random measure zeta of a stationary process is defined on sets E subset of (-infinity, infinity) x (0, infinity), as the time which the process (suitably normalized) spends in the set E. Particular cases thus include a multitude of features (including sojourn times) related to high levels. It is therefore not surprising that a single limit theorem for zeta at high levels contains a wide variety of useful extremal and high level exceedance results for the stationary process itself. The theory given for the excursion random measure demonstrates, under very general conditions, its asymptotic infinite divisibility with certain stability and independence of increments properties leading to its asymptotic distribution (Theorem 4.1). The results are illustrated by a number of examples including stable and Gaussian processes.