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Renewal in Hawkes processes with self-excitation and inhibition

Manon Costa, Carl Graham, Laurence Marsalle, Viet Chi Tran

TL;DR

This work analyzes Hawkes processes on $[0,\infty)$ with a signed reproduction function $h$ to model both self-excitation and inhibition. Owing to inhibition, the classical cluster representation breaks down, so the authors develop a renewal-theoretic framework built on a strong Markov process $X_t=(S_t N^h)|_{(-A,0]}$ and a coupling to the nonnegative case via $h^+$, enabling a regenerative decomposition. They prove an ergodic theorem and a central limit theorem for sliding-window functionals, together with non-asymptotic exponential concentration bounds; exponential moments for renewal times are established through a $M/G/\infty$-queue analysis. These results extend prior concentration and ergodic results to Hawkes processes with inhibition and provide tools for statistical inference in systems exhibiting both excitation and inhibition, with independent interest in renewal theory and queueing theory.

Abstract

This paper investigates Hawkes processes on the positive real line exhibiting both self-excitation and inhibition. Each point of this point process impacts its future intensity by the addition of a signed reproduction function. The case of a nonnegative reproduction function corresponds to self-excitation, and has been widely investigated in the literature. In particular, there exists a cluster representation of the Hawkes process which allows to apply results known for Galton-Watson trees. In the present paper, we establish limit theorems for Hawkes process with signed reproduction functions by using renewal techniques. We notably prove exponential concentration inequalities, and thus extend results of Reynaud-Bouret and Roy (2007) which were proved for nonnegative reproduction functions using this cluster representation which is no longer valid in our case. An important step for this is to establish the existence of exponential moments for renewal times of M/G/infinity queues that appear naturally in our problem. These results have their own interest, independently of the original problem for the Hawkes processes.

Renewal in Hawkes processes with self-excitation and inhibition

TL;DR

This work analyzes Hawkes processes on with a signed reproduction function to model both self-excitation and inhibition. Owing to inhibition, the classical cluster representation breaks down, so the authors develop a renewal-theoretic framework built on a strong Markov process and a coupling to the nonnegative case via , enabling a regenerative decomposition. They prove an ergodic theorem and a central limit theorem for sliding-window functionals, together with non-asymptotic exponential concentration bounds; exponential moments for renewal times are established through a -queue analysis. These results extend prior concentration and ergodic results to Hawkes processes with inhibition and provide tools for statistical inference in systems exhibiting both excitation and inhibition, with independent interest in renewal theory and queueing theory.

Abstract

This paper investigates Hawkes processes on the positive real line exhibiting both self-excitation and inhibition. Each point of this point process impacts its future intensity by the addition of a signed reproduction function. The case of a nonnegative reproduction function corresponds to self-excitation, and has been widely investigated in the literature. In particular, there exists a cluster representation of the Hawkes process which allows to apply results known for Galton-Watson trees. In the present paper, we establish limit theorems for Hawkes process with signed reproduction functions by using renewal techniques. We notably prove exponential concentration inequalities, and thus extend results of Reynaud-Bouret and Roy (2007) which were proved for nonnegative reproduction functions using this cluster representation which is no longer valid in our case. An important step for this is to establish the existence of exponential moments for renewal times of M/G/infinity queues that appear naturally in our problem. These results have their own interest, independently of the original problem for the Hawkes processes.

Paper Structure

This paper contains 21 sections, 18 theorems, 168 equations, 2 figures.

Table of Contents

  1. Introduction and main results
  2. Definitions and notations
  3. Main Results
  4. Hawkes processes
  5. Solving the equation for the Hawkes process
  6. The cluster representation for nonnegative reproduction functions
  7. An auxiliary Markov Process
  8. Definition of a strong Markov process
  9. Renewal of $X$ at $\emptyset$
  10. Step 1: Analysis of the problem.
  11. Step 2: Exponential moments of $E$.
  12. Step 3: Estimate of the tail distribution of $\tau^+$.
  13. Proof of the main results
  14. Appendix
  15. Proof of Proposition \ref{['prop:couplage']}
  16. ...and 6 more sections

Key Result

Theorem 1.3

Let $N^h$ be a Hawkes process with immigration rate $\lambda >0$, reproduction function $h: (0,+\infty) \to \mathbb{R}$, and initial condition $N^0$ with law $\mathfrak{m}$, satisfying Assumption hyp:h. Let $A<\infty$ be such that $A\ge L(h)$, and $\pi_A$ be the probability measure on $\mathcal{N}((

Figures (2)

  • Figure 2.1: (a) Hawkes process with a positive reproduction function $h$. (b) Hawkes process with a general reproduction function $h$. The dots in the plane represent the atoms of the Poisson point process $Q$ used for the construction. The atoms of the Hawkes processes are the green dots on the abscissa axis. The bold red curve corresponds to the intensity $\Lambda^h$ and the colored curves represent the partial cumulative contributions of the successive atoms of the Hawkes process. In (b), the bold blue curve corresponds to the intensity of the dominating Hawkes process with reproduction function $h^+$.
  • Figure 2.2: Cluster representation of a Hawkes process with positive reproduction function. The abscissa of the dots give its atoms. Offspring are colored according to their ancestor, and their ordinates correspond to their generation in this age-structured Galton-Watson tree.

Theorems & Definitions (36)

  • Definition 1.1
  • Theorem 1.3: Ergodic theorems
  • Theorem 1.4: Central limit theorem
  • Theorem 1.5: Concentration inequalities
  • Corollary 1.6
  • Remark 1.7
  • Remark 1.8
  • Proposition 2.1
  • Remark 2.2
  • Remark 2.3
  • ...and 26 more