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A functional Breuer-Major theorem with Poisson noise

Fanhao Kong, Haiyi Wang

Abstract

We extend the functional Breuer-Major theorem for Gaussians to the Poisson case, where the stationary sequence arises from a Poisson point process. We use the $L^p$ spectral gap inequality of Poisson point process as a tool to prove tightness.

A functional Breuer-Major theorem with Poisson noise

Abstract

We extend the functional Breuer-Major theorem for Gaussians to the Poisson case, where the stationary sequence arises from a Poisson point process. We use the spectral gap inequality of Poisson point process as a tool to prove tightness.

Paper Structure

This paper contains 13 sections, 36 theorems, 168 equations.

Table of Contents

  1. Introduction
  2. The Breuer-Major theorem for Gaussians
  3. Main result
  4. Preliminaries
  5. Malliavin calculus for Poisson point process
  6. Local bounds of widehat phi
  7. Tightness
  8. Tightness of (mathcalT ge d Y n) n geq 1
  9. Tightness of (mathcalT ge d Z n) n geq 1
  10. Convergence of finite-dimensional distributions
  11. A formula for expectation of product of Wiener-Itô integrals
  12. Proof of Theorem \ref{['thm:fdd_convergence']}
  13. Convergence of bBn, ell,m

Key Result

Theorem 1.2

Fix $d\in\mathbb{N}^+$, $\alpha > \frac{1}{2} + \frac{1}{2d}$ and $\gamma_0>d_\alpha+1$. Suppose that there exist $C_\psi>0$ and $M_0\in \mathbb{N}$ such that for all $x\in\mathbb{R}$ and Then, the process $\mathcal{T}^{\ge d} Y_n$ converges in law to $\mu B$ in the Skorohod space $\mathbf{D}([0,1])$, where $(B_t)_{t \in [0,1]}$ is a standard Brownian motion, and

Theorems & Definitions (70)

  • Definition 1.1
  • Theorem 1.2
  • proof
  • Remark 1.3
  • Remark 1.4
  • Theorem 1.5
  • proof
  • Lemma 2.1
  • Lemma 2.2
  • proof
  • ...and 60 more