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Tail asymptotics and precise large deviations for some Poisson cluster processes

Fabien Baeriswyl, Valérie Chavez-Demoulin, Olivier Wintenberger

TL;DR

This work analyzes tail behavior of maxima and sums of transformed marks in two clustered point-process models, the renewal Poisson cluster process and the Hawkes process. By assuming regular variation for the driving components and applying Karamata's Tauberian theorem, it derives explicit tail asymptotics for cluster functionals $H^R$, $D^R$, $H^H$, and $D^H$, and translates these into precise large deviation principles for the processes over finite horizons. The main contributions are exact tail forms for each functional (eg, $P(H^R>x)\sim(1+\mathbb{E}[K_A])P(X>x)$ and $P(D^R>x)\sim P(X+\mathbb{E}[X]K_A>x)+\mathbb{E}[K_A]P(X>x)$ in renewal, and analogous results in Hawkes) and the corresponding sharp large-deviation results on $[0,T]$ with explicit rates governed by cluster counts and subcriticality conditions. These results enable sharp tail risk assessments in applications with clustering such as seismology, insurance, and meteorology by providing finite-horizon tail probabilities for both maxima and sums of cluster functionals.

Abstract

We study the tail asymptotics of two functionals (the maximum and the sum of the marks) of a generic cluster in two sub-models of the marked Poisson cluster process, namely the renewal Poisson cluster process and the Hawkes process. Under the hypothesis that the governing components of the processes are regularly varying, we extend results due to [18] and [5] notably, relying on Karamata's Tauberian Theorem to do so. We use these asymptotics to derive precise large deviation results in the fashion of [30] for the above-mentioned processes.

Tail asymptotics and precise large deviations for some Poisson cluster processes

TL;DR

This work analyzes tail behavior of maxima and sums of transformed marks in two clustered point-process models, the renewal Poisson cluster process and the Hawkes process. By assuming regular variation for the driving components and applying Karamata's Tauberian theorem, it derives explicit tail asymptotics for cluster functionals , , , and , and translates these into precise large deviation principles for the processes over finite horizons. The main contributions are exact tail forms for each functional (eg, and in renewal, and analogous results in Hawkes) and the corresponding sharp large-deviation results on with explicit rates governed by cluster counts and subcriticality conditions. These results enable sharp tail risk assessments in applications with clustering such as seismology, insurance, and meteorology by providing finite-horizon tail probabilities for both maxima and sums of cluster functionals.

Abstract

We study the tail asymptotics of two functionals (the maximum and the sum of the marks) of a generic cluster in two sub-models of the marked Poisson cluster process, namely the renewal Poisson cluster process and the Hawkes process. Under the hypothesis that the governing components of the processes are regularly varying, we extend results due to [18] and [5] notably, relying on Karamata's Tauberian Theorem to do so. We use these asymptotics to derive precise large deviation results in the fashion of [30] for the above-mentioned processes.
Paper Structure (13 sections, 12 theorems, 38 equations)

This paper contains 13 sections, 12 theorems, 38 equations.

Table of Contents

  1. Introduction
  2. Random functionals of clusters
  3. Mixed binomial Poisson cluster process
  4. Renewal Poisson cluster process
  5. Hawkes process
  6. A word on regular variation
  7. Tail asymptotics of maximum functional in renewal Poisson cluster process
  8. Tail asymptotics of the sum functional in renewal Poisson cluster process
  9. Tail asymptotics of the maximum functional in the Hawkes process
  10. Tail asymptotics of the sum functional in Hawkes process
  11. Precise large deviations of cluster process functionals
  12. Large deviations of maxima over an interval $[0,T]$
  13. Large deviations of sums over an interval $[0,T]$

Key Result

Proposition 3.1

A random vector $\mathbf{X}$ with values in $\mathbb{R}^d$ is regularly varying with index $\alpha >0$ and non-null Radon measure $\mu$ on $\mathbb{R}^d_{\mathbf{0}}$ if and only if one of the following relations holds:

Theorems & Definitions (22)

  • Remark 2.1
  • Proposition 3.1: Theorem 6.1 in resnick2
  • Proposition 3.2: Theorem 2.2.30 in mw23, Proposition 4.3 and Corollary 4.2 in lrr14
  • Example 3.1
  • Proposition 3.3: Proposition 3.1 in szm16
  • Theorem 3.1: Karamata's Tauberian Theorem, Theorem 8.1.6 in bgt89
  • Remark 3.1
  • Proposition 4.1
  • Remark 4.1
  • Proposition 5.1
  • ...and 12 more