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Exponential moments for Hawkes processes under minimal assumptions

Théo Leblanc

TL;DR

The paper proves that under the minimal condition for the existence of a stationary Hawkes process—namely that the spectral radius of the matrix of $L^1$-norms of the interaction functions is below one—the count of points in any bounded interval possesses exponential moments. The authors derive explicit, parameter-dependent bounds by combining a cluster representation with a mass transport principle and Galton–Watson branching arguments, obtaining sharp conditions in terms of a Geometric bound $\bm{H}\in\bm{\mathop{Ge}}(r,K)$. A functional extension is provided, showing exponential moments for counts against nonnegative test functions $f$ with bounds depending on a discretized $L^1$-norm $|f|^{\infty}_{1,T}$. These results tighten the understanding of moment properties of Hawkes processes and facilitate robust statistical analyses under the minimal stationarity condition.

Abstract

We prove that the number of points of a stationary linear Hawkes process lying in any bounded subset of the real line has exponential moments, without any other assumption than the one needed for existence of such stationary process, namely the spectral radius of the matrix of L1 norms of interaction functions is smaller than one. The proof relies on a mass transport principle argument. We also specify the dependence of the bounds with respect to the base rates and the matrix of L1 norms of interaction functions defining the Hawkes process and give a functional version of the result.

Exponential moments for Hawkes processes under minimal assumptions

TL;DR

The paper proves that under the minimal condition for the existence of a stationary Hawkes process—namely that the spectral radius of the matrix of -norms of the interaction functions is below one—the count of points in any bounded interval possesses exponential moments. The authors derive explicit, parameter-dependent bounds by combining a cluster representation with a mass transport principle and Galton–Watson branching arguments, obtaining sharp conditions in terms of a Geometric bound . A functional extension is provided, showing exponential moments for counts against nonnegative test functions with bounds depending on a discretized -norm . These results tighten the understanding of moment properties of Hawkes processes and facilitate robust statistical analyses under the minimal stationarity condition.

Abstract

We prove that the number of points of a stationary linear Hawkes process lying in any bounded subset of the real line has exponential moments, without any other assumption than the one needed for existence of such stationary process, namely the spectral radius of the matrix of L1 norms of interaction functions is smaller than one. The proof relies on a mass transport principle argument. We also specify the dependence of the bounds with respect to the base rates and the matrix of L1 norms of interaction functions defining the Hawkes process and give a functional version of the result.

Paper Structure

This paper contains 9 sections, 5 theorems, 48 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Motivation
  3. Setting
  4. Main result
  5. Functional result
  6. Cluster representation
  7. Proof of \ref{['moment exp']}
  8. Proof of \ref{['moment exp GW']}
  9. Proof of \ref{['functional']}

Key Result

Theorem 1

Suppose that A1 holds, ie the spectral radius of $\bm{H} = (\Vert h^m_{m'} \Vert_1)_{m',m}$ is strictly less than $1$. Then there exists $\xi>0$ depending only on $\bm{H}$ such that for all $B\subset \mathds{R}$ bounded, we have More precisely, if $B\subset [s,s+L)$ for some $s\in\mathds{R}$ and $L>0$, and if $\bm{H}\in\bm{\mathop{\mathrm{Ge}}\nolimits(} r,K \bm{)}$ with $0< r< 1$ and $0\leq K<\i

Figures (2)

  • Figure 1: Illustration of the cluster representation. The red dots are the immigrant points. They arrive according to the immigrant rate, being $\mu_m$ for type $m\in\bm{M}$, and give birth to clusters, represented by the blue shapes. In the end, only the birth dates and the types are relevant for Hawkes processes, which can be interpreted as a projection of the blue shapes on the time axis, in the sense that the tree structure is forgotten
  • Figure 2: Illustration of the mass transport principle. In expectation, both side are equal.

Theorems & Definitions (10)

  • Definition 1
  • Definition 2
  • Theorem 1
  • Remark 1
  • Theorem 2
  • Proposition 1
  • Lemma 1
  • proof : Proof of \ref{['repartition of a cluster']}
  • Lemma 2
  • proof : Proof of \ref{['moment exp GW']}