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Non-central limit of densities of some functionals of Gaussian processes

Solesne Bourguin, Thanh Dang, Yaozhong Hu

TL;DR

The paper develops a density-level Gamma-approximation theory for Gaussian functionals, introducing a novel density representation formula in the Markov-diffusion/Malliavin framework and applying it to Gamma targets. It extends the Malliavin-Stein method to non-central limits, delivering explicit, moment-based density bounds for functionals living in a fixed even Wiener chaos as well as for general Gaussian functionals with infinite chaos expansions. A key contribution is the Gamma-density representation for Laguerre-type generators and the accompanying derivative estimates, enabling precise control of p_F(x) and its derivatives relative to the Gamma density p_{\mathcal{G}_\alpha}(x). The results cover an application to random variables in the second Wiener chaos, where Gamma-structure elements are known to appear, and provide usable quantitative benchmarks for density convergence in non-Gaussian limit regimes with potential statistical implications.

Abstract

We establish the convergence of the densities of a sequence of nonlinear functionals of an underlying Gaussian process to the density of a Gamma distribution. The key idea of our work is a new density formula for random variables in the setting of Markov diffusion generators, which yields a special representation for the density of a Gamma distribution. Via this representation, we are able to provide precise estimates on the distance between densities while developing the techniques of Malliavin calculus and Stein's method suitable to Gamma approximation at the density level. We first focus our study on the case of random variables living in a fixed Wiener chaos of an even order for which the bound for the difference of the densities can be dominated by a linear combination of moments up to order four. We then study the case of general Gaussian functionals with possibly infinite chaos expansion. Finally, we provide an application to random variables living in the second Wiener chaos.

Non-central limit of densities of some functionals of Gaussian processes

TL;DR

The paper develops a density-level Gamma-approximation theory for Gaussian functionals, introducing a novel density representation formula in the Markov-diffusion/Malliavin framework and applying it to Gamma targets. It extends the Malliavin-Stein method to non-central limits, delivering explicit, moment-based density bounds for functionals living in a fixed even Wiener chaos as well as for general Gaussian functionals with infinite chaos expansions. A key contribution is the Gamma-density representation for Laguerre-type generators and the accompanying derivative estimates, enabling precise control of p_F(x) and its derivatives relative to the Gamma density p_{\mathcal{G}_\alpha}(x). The results cover an application to random variables in the second Wiener chaos, where Gamma-structure elements are known to appear, and provide usable quantitative benchmarks for density convergence in non-Gaussian limit regimes with potential statistical implications.

Abstract

We establish the convergence of the densities of a sequence of nonlinear functionals of an underlying Gaussian process to the density of a Gamma distribution. The key idea of our work is a new density formula for random variables in the setting of Markov diffusion generators, which yields a special representation for the density of a Gamma distribution. Via this representation, we are able to provide precise estimates on the distance between densities while developing the techniques of Malliavin calculus and Stein's method suitable to Gamma approximation at the density level. We first focus our study on the case of random variables living in a fixed Wiener chaos of an even order for which the bound for the difference of the densities can be dominated by a linear combination of moments up to order four. We then study the case of general Gaussian functionals with possibly infinite chaos expansion. Finally, we provide an application to random variables living in the second Wiener chaos.
Paper Structure (13 sections, 21 theorems, 210 equations)

This paper contains 13 sections, 21 theorems, 210 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Markov diffusion generators
  4. Malliavin calculus on Gaussian space
  5. Density representation formulas
  6. Density representation formula for the Gamma distribution
  7. Density representation for multiple Wiener integrals
  8. Stein's method
  9. Random variables in a fixed Wiener chaos of even order
  10. Pointwise estimates for densities
  11. Pointwise estimate for derivatives of densities
  12. General Gaussian functionals
  13. Application to random variables in the second Wiener chaos

Key Result

Theorem 1.1

Let $F$ be a multiple Wiener integral of an even order $q\geq 2$ such that $\mathbb{E}{\left[ F^2\right]}=\alpha$. Denote by $p_{F+\alpha}$ the density function of $F +\alpha$. Then, for every $x \neq 0$ and $\alpha >0$, one has the pointwise density estimate where $P_0(x)$ is a positive quantity depending on $x$ given by where the factors $d_1(x),d_2(x),d_3(x)$ are positive and finite for every

Theorems & Definitions (49)

  • Theorem 1.1
  • Theorem 1.2
  • Example 2.1
  • Lemma 2.2
  • Lemma 2.3
  • Proposition 3.1
  • proof
  • Proposition 3.2
  • proof
  • Remark 3.3
  • ...and 39 more