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Gaussian Approximation and Moderate Deviations of Poisson Shot Noises with Application to Compound Generalized Hawkes Processes

Mahmoud Khabou, Giovanni Luca Torrisi

TL;DR

This work develops quantitative Gaussian approximation and moderate deviation theory for first-chaos functionals on Poisson spaces, and applies it to Poisson shot-noise models, including generalized compound Hawkes processes. By leveraging the Last–Peccati–Schulte framework and cumulant control, it provides explicit Wasserstein and Kolmogorov distance bounds, Bernstein-type concentration inequalities, and Normal approximations with Cramér corrections. The results are then specialized to spatial Poisson shot-noise, compound Poisson cluster processes, generalized Hawkes processes with Poisson or Binomial offspring, and wireless interference models, yielding practical, explicit performance guarantees. The methodology broadens the Hawkes literature to spatial settings, offering tractable, applicability-oriented bounds with potential impact in communications, risk, and spatial statistics.

Abstract

In this article, we give explicit bounds on the Wasserstein and the Kolmogorov distances between random variables lying in the first chaos of the Poisson space and the standard Normal distribution, using the results proved by Last, Peccati and Schulte. Relying on the theory developed in the work of Saulis and Statulevicius and on a fine control of the cumulants of the first chaoses, we also derive moderate deviation principles, Bernstein-type concentration inequalities and Normal approximation bounds with Cramér correction terms for the same variables. The aforementioned results are then applied to Poisson shot-noise processes and, in particular, to the generalized compound Hawkes point processes (a class of stochastic models, introduced in this paper, which generalizes classical Hawkes processes). This extends the recent results availale in the literature regarding the Normal approximation and moderate deviations.

Gaussian Approximation and Moderate Deviations of Poisson Shot Noises with Application to Compound Generalized Hawkes Processes

TL;DR

This work develops quantitative Gaussian approximation and moderate deviation theory for first-chaos functionals on Poisson spaces, and applies it to Poisson shot-noise models, including generalized compound Hawkes processes. By leveraging the Last–Peccati–Schulte framework and cumulant control, it provides explicit Wasserstein and Kolmogorov distance bounds, Bernstein-type concentration inequalities, and Normal approximations with Cramér corrections. The results are then specialized to spatial Poisson shot-noise, compound Poisson cluster processes, generalized Hawkes processes with Poisson or Binomial offspring, and wireless interference models, yielding practical, explicit performance guarantees. The methodology broadens the Hawkes literature to spatial settings, offering tractable, applicability-oriented bounds with potential impact in communications, risk, and spatial statistics.

Abstract

In this article, we give explicit bounds on the Wasserstein and the Kolmogorov distances between random variables lying in the first chaos of the Poisson space and the standard Normal distribution, using the results proved by Last, Peccati and Schulte. Relying on the theory developed in the work of Saulis and Statulevicius and on a fine control of the cumulants of the first chaoses, we also derive moderate deviation principles, Bernstein-type concentration inequalities and Normal approximation bounds with Cramér correction terms for the same variables. The aforementioned results are then applied to Poisson shot-noise processes and, in particular, to the generalized compound Hawkes point processes (a class of stochastic models, introduced in this paper, which generalizes classical Hawkes processes). This extends the recent results availale in the literature regarding the Normal approximation and moderate deviations.
Paper Structure (35 sections, 17 theorems, 196 equations)

This paper contains 35 sections, 17 theorems, 196 equations.

Table of Contents

  1. Introduction
  2. Poisson shot noise random variables
  3. Compound Poisson cluster point processes
  4. Generalized Hawkes processes and generalized compound Hawkes processes
  5. Interference in wireless communication
  6. Informal description of the results
  7. Gaussian approximation and moderate deviations of the first chaos on the Poisson space
  8. Bounds on the Wasserstein and Kolmogorov distances between the law of a first chaos on the Poisson space and the standard Normal distribution
  9. Moderate deviations, Bernstein-type concentration inequalities and Normal approximation bounds with Cramér correction term for first chaoses on the Poisson space
  10. Application to Poisson shot noise random variables
  11. Gaussian approximation
  12. Moderate deviations, Bernstein-type concentration inequalities and Normal approximation bounds with Cramér correction term
  13. Application to a class of compound Poisson cluster point processes
  14. Gaussian approximation
  15. Moderate deviations, Bernstein-type concentration inequalities and Normal approximation bounds with Cramér correction term
  16. ...and 20 more sections

Key Result

Theorem 4.1

Let $f\in L^2(\alpha)$ be such that $\|f\|_{L^2(\alpha)}=1$. Then and

Theorems & Definitions (23)

  • Theorem 4.1
  • Definition 4.2
  • Theorem 4.3
  • Theorem 5.1
  • Remark 5.2
  • Theorem 5.3
  • Remark 5.4
  • Corollary 6.1
  • Lemma 6.2
  • Corollary 6.3
  • ...and 13 more