Hidden regular variation for point processes and the single/multiple large point heuristic

TL;DR

This paper investigates the asymptotic behavior of marked point processes with heavy-tailed marks, revealing a hierarchy of hidden regular variation phenomena and applying these insights to risk models in reinsurance.

Contribution

It introduces the single and multiple large point heuristics for regular variation in point processes and extends hidden regular variation analysis to the Skorokhod space with applications in risk theory.

Findings

Limit measure concentrates on single large points.

Successive hidden regular variation reveals multiple large points.

Application to reinsurance shows asymptotic residual risk behavior.

Abstract

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, $k\geq 1$, 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. 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.