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Stable limit theorems on the Poisson space

Ronan Herry

Abstract

We prove limit theorems for functionals of a Poisson point process using the Malliavin calculus on the Poisson space. The target distribution is conditionally either a Gaussian vector or a Poisson random variable. The convergence is stable and our conditions are expressed in terms of the Malliavin operators. For conditionally Gaussian limits, we also obtain quantitative bounds, given for the Monge-Kantorovich transport distance in the univariate case; and for another probabilistic variational distance in higher dimension. Our work generalizes several limit theorems on the Poisson space, including the seminal works by Peccati, Solé, Taqqu & Utzet for Gaussian approximations; and by Peccati for Poisson approximations; as well as the recently established fourth-moment theorem on the Poisson space of Döbler & Peccati. We give an application to stochastic processes.

Stable limit theorems on the Poisson space

Abstract

We prove limit theorems for functionals of a Poisson point process using the Malliavin calculus on the Poisson space. The target distribution is conditionally either a Gaussian vector or a Poisson random variable. The convergence is stable and our conditions are expressed in terms of the Malliavin operators. For conditionally Gaussian limits, we also obtain quantitative bounds, given for the Monge-Kantorovich transport distance in the univariate case; and for another probabilistic variational distance in higher dimension. Our work generalizes several limit theorems on the Poisson space, including the seminal works by Peccati, Solé, Taqqu & Utzet for Gaussian approximations; and by Peccati for Poisson approximations; as well as the recently established fourth-moment theorem on the Poisson space of Döbler & Peccati. We give an application to stochastic processes.

Paper Structure

This paper contains 24 sections, 21 theorems, 142 equations.

Table of Contents

  1. Preliminaries
  2. Notations
  3. Probabilistic approximations and limit theorems
  4. Definition of Poisson point processes
  5. Gaussian and Poisson mixtures
  6. Stochastic analysis for Poisson point processes
  7. Main abstract results
  8. Main qualitative results
  9. Convergence to a Gaussian mixture
  10. Convergence to a Poisson mixture
  11. Main quantitative results for Gaussian mixtures
  12. General results in any dimension
  13. Bounds in the Wasserstein distance for the one-dimensional case
  14. Comparison with existing results
  15. Proofs
  16. ...and 9 more sections

Key Result

Theorem \oldthetheorem

Let the previous notation prevails, and assume that: and ThenTo be precise, the theorem of PeccatiSoleTaqquUtzet chooses one particular solution of $F_{n} = \delta u_{n}$ but we do not enter into too many technical details in this introduction., we have that $F_{n} \xrightarrow[{n}\to{\infty}]{law} \mathbf{N}(0,\sigma^{2})$.

Theorems & Definitions (46)

  • Theorem \oldthetheorem: PeccatiSoleTaqquUtzet
  • Proposition \oldthetheorem
  • proof
  • Lemma \oldthetheorem
  • proof
  • Proposition \oldthetheorem
  • Theorem \oldthetheorem
  • Remark 1
  • Theorem \oldthetheorem
  • Remark 2
  • ...and 36 more