2022/07/22 by Anja Janßen, Johan Segers, Janßen, Anja +1
Economics, Econometrics and Finance · #60G10 #60G70 #FOS: Mathematics #Financial Risk and Volatility Modeling #Market Dynamics and Volatility #Probability (math.PR)
paper · pdf · doi:10.48550/arxiv.2207.11204
openalex publication_date 2022/07/22 · openalex created_date 2022/07/27 · openalex updated_date 2026/07/28
Motivated by examples from extreme value theory we introduce the general notion of a cluster process as a limiting point process of returns of a certain event in a time series. We explore general invariance properties of cluster processes which are implied by stationarity of the underlying time series under minimal assumptions. Of particular interest are the cluster size distributions, where we introduce the two notions of inspected and typical cluster sizes and derive general properties of and connections between them. While the extremal index commonly used in extreme value theory is often interpreted as the inverse of a "mean cluster size", we point out that this only holds true for the expected value of the typical cluster size, caused by an effect very similar to the inspection paradox in renewal theory.