[Paper Review] Fourth Moment Theorems for complex Gaussian approximation

Simon Campese|arXiv (Cornell University)|Nov 2, 2015

Stochastic processes and financial applications17 references10 citations

TL;DR

This paper establishes a quantitative Fourth Moment Theorem for complex Gaussian approximation using Stein's method and a complex extension of Γ-calculus within the framework of complex Markov diffusion generators. It provides a Wasserstein distance bound between vectors of smooth complex random variables and complex Gaussians, with the bound expressible in terms of second and fourth absolute moments for chaotic eigenfunctions, thus generalizing real-valued results to the complex domain with enhanced analytic tools via Wirtinger derivatives and holomorphic structures.

ABSTRACT

We prove a bound for the Wasserstein distance between vectors of smooth complex random variables and complex Gaussians in the framework of complex Markov diffusion generators. For the special case of chaotic eigenfunctions, this bound can be expressed in terms of certain fourth moments of the vector, yielding a quantitative Fourth Moment Theorem for complex Gaussian approximation on complex Markov diffusion chaos. This extends results of Azmoodeh, Campese, Poly (2014) and Campese, Nourdin, Peccati (2015) for the real case. Our main ingredients are a complex version of the so called $Γ$-calculus and Stein's method for the multivariate complex Gaussian distribution.

Motivation & Objective

• To extend the abstract Fourth Moment Theorem framework—previously developed for real-valued random variables in the context of Markov diffusion generators—to the complex domain.
• To establish a quantitative bound on the Wasserstein distance between vectors of smooth complex random variables and complex Gaussian vectors.
• To develop a complex version of Γ-calculus based on Wirtinger derivatives ∂z and ∂z̄ to handle complex-valued Malliavin-type calculus.
• To derive a complex counterpart of the real multidimensional Fourth Moment Theorem, particularly for chaotic eigenfunctions of complex Markov generators.
• To demonstrate that convergence in distribution to a complex Gaussian is characterized by convergence of second and fourth absolute moments for such chaotic vectors.

Proposed method

• Adapt Stein’s method to the multivariate complex Gaussian distribution using complex-valued test functions and the complex Stein equation.
• Introduce a complex version of the carré du champ operator Γ and its iterated form Γ₂, defined via Wirtinger derivatives ∂z and ∂z̄.
• Employ the complex Ornstein-Uhlenbeck generator as a key example, where eigenfunctions correspond to complex multiple Wiener-Itô integrals.
• Derive a Wasserstein distance bound involving the fourth moments of the vector and its interaction with the inverse generator L⁻¹ via integration by parts.
• Use the complex Γ-calculus to control the error in Stein’s method, leveraging holomorphicity and unitary invariance structures absent in the real case.
• Apply the Gaussian integration-by-parts formula in the complex setting to relate moments of the target complex Gaussian to the moments of the random vector.

Experimental results

Research questions

• RQ1 Can the Fourth Moment Theorem for real Gaussian approximation be extended to the complex domain using abstract diffusion generator theory?
• RQ2 How can Stein’s method be adapted to provide quantitative bounds for the multivariate complex Gaussian distribution?
• RQ3 What is the role of complex analysis tools—such as Wirtinger derivatives and holomorphic functions—in simplifying or strengthening the bounds in the complex setting?
• RQ4 For chaotic eigenfunctions of complex Markov generators, is convergence in distribution to a complex Gaussian fully characterized by the convergence of second and fourth absolute moments?
• RQ5 Can the abstract framework of Markov diffusion generators be extended to include complex-valued processes while preserving the key tools of Γ-calculus and Malliavin-type integration by parts?

Key findings

• A quantitative bound on the Wasserstein distance between a vector of smooth complex random variables and a complex Gaussian is established using Stein’s method and complex Γ-calculus.
• For chaotic eigenfunctions of a complex Markov diffusion generator, the Wasserstein distance is bounded by a constant multiple of the sum of the fourth cumulants and the squared L²-norms of the L⁻¹F_k terms.
• The bound simplifies to a function of second and fourth absolute moments for chaotic vectors, yielding a complex Fourth Moment Theorem: convergence in distribution to a complex Gaussian is equivalent to convergence of these moments.
• The result generalizes [ACP14] and [CNPP15] to the complex case, with Corollary 4.7 showing that for d=1, convergence holds iff the second and fourth absolute moments converge to σ² and 2σ⁴, respectively.
• For d≥2, the complex Peccati-Tudor Theorem (Proposition 4.9) establishes equivalence between joint convergence and marginal convergence under ergodicity and spectral separation conditions.
• The complex framework reveals additional structure—such as unitary group invariance and holomorphic decomposition in eigenspaces—unavailable in the real case, which may aid future theoretical developments.