A note on the point processes of rare events

Abstract

We discuss the limits of point processes which are generated by a triangular array of rare events. Such point processes are motivated by the exceedances of a high boundary by a random sequence since exceedances are rare events in this case. This application relates the problem to extreme value theory from where the method is used to treat the asymptotic approximation of these point processes. The presented general approach extends, unifies and clarifies some of the various conditions used in the extreme value theory.

Rare events, temporal dependence, and the extremal index

“…These complement the assessment of the accuracy of the compound Poisson approximation for the empirical point process of exceedances in Barbour et al (2002). Furthermore, these inequalities lead to asymptotic results that serve on the one hand to formulate, in the framework of rare events, known characterizations of the extremal index (Leadbetter (1983), O'Brien (1987), Ferro and Segers (2003)), and on the other hand to complement various Poisson limit results for triangular arrays (Hüsler (1993), Hüsler and Schmidt (1996)). Point process results will not be pursued in this paper, as the dependence restrictions in force will be weaker than those in the aforementioned papers.…”

Rare Events, Temporal Dependence, and the Extremal Index

“…These complement the assessment of the accuracy of the compound Poisson approximation for the empirical point process of exceedances in Barbour, Xia and Novak (2002). Further, these inequalities lead to asymptotic results, which serve on the one hand to formulate in the framework of rare events known characterizations of the extremal index (Leadbetter, 1983;O'Brien, 1987;Ferro and Segers, 2003), and on the other hand to complement various Poisson limit results for triangular arrays (Hüsler, 1993;Hüsler and Schmidt, 1996). Point process results will not be pursued in this paper as the dependence restrictions in force will be weaker than in the aforementioned papers.…”

Rare events, temporal dependence, and the extremal index

“…These complement the assessment of the accuracy of the compound Poisson approximation for the empirical point process of exceedances in Barbour, Xia and Novak (2002). Further, these inequalities lead to asymptotic results, which serve on the one hand to formulate in the framework of rare events known characterizations of the extremal index (Leadbetter, 1983;O'Brien, 1987;Ferro and Segers, 2003), and on the other hand to complement various Poisson limit results for triangular arrays (Hüsler, 1993;Hüsler and Schmidt, 1996). Point process results will not be pursued in this paper as the dependence restrictions in force will be weaker than in the aforementioned papers.…”