[Paper Review] Malliavin-Stein method for Variance-Gamma approximation on Wiener space

Peter Eichelsbacher, Christoph Thäle|arXiv (Cornell University)|Sep 19, 2014

Random Matrices and Applications20 references4 citations

TL;DR

This paper develops a Malliavin-Stein method for Variance-Gamma approximation on Wiener space, combining Malliavin calculus with Stein's method to derive explicit bounds for the convergence of chaotic random variables to a Variance-Gamma distribution. The key result is a six moment theorem: a sequence in the second Wiener chaos converges to a Variance-Gamma distribution if and only if its first six moments converge to those of the target distribution.

ABSTRACT

We combine Malliavin calculus with Stein's method to derive bounds for the Variance-Gamma approximation of functionals of isonormal Gaussian processes, in particular of random variables living inside a fixed Wiener chaos induced by such a process. The bounds are presented in terms of Malliavin operators and norms of contractions. We show that a sequence of distributions of random variables in the second Wiener chaos converges to a Variance-Gamma distribution if and only if their moments of order two to six converge to that of a Variance-Gamma distributed random variable (six moment theorem). Moreover, simplified versions for Laplace or symmetrized Gamma distributions are presented. Also multivariate extensions and a universality result for homogeneous sums are considered.

Motivation & Objective

• To extend the Malliavin-Stein method beyond Gaussian and Gamma approximations to the broader class of Variance-Gamma distributions.
• To establish quantitative bounds for the convergence of random variables in a fixed Wiener chaos to a Variance-Gamma distribution.
• To prove a six moment theorem characterizing convergence in distribution to a Variance-Gamma law via convergence of moments up to order six.
• To generalize results to multivariate settings and establish a universality result for homogeneous sums in the context of Variance-Gamma approximation.

Proposed method

• Combines Malliavin calculus with Stein's method to derive bounds on the distance between chaotic functionals and a Variance-Gamma distribution.
• Uses Malliavin operators and norms of contractions to express convergence rates in terms of cumulants and higher-order moments.
• Applies integration-by-parts formulas and Stein equation solutions for non-symmetric Variance-Gamma distributions via smooth test functions.
• Employs the structure of Wiener chaos and isonormal Gaussian processes to analyze convergence in the second chaos and beyond.
• Derives bounds involving terms like $ A_n(j) $ and $ B_n(i,j) $, which are linked to Malliavin norms and covariances of squared components.
• Utilizes the strong asymptotic independence of Wiener chaos components to show joint convergence under moment and covariance conditions.

Experimental results

Research questions

• RQ1 Under what conditions does a sequence of random variables in the second Wiener chaos converge in distribution to a Variance-Gamma distribution?
• RQ2 Can explicit bounds on the approximation error be derived using Malliavin calculus and Stein's method for Variance-Gamma targets?
• RQ3 Is there a six moment theorem for Variance-Gamma approximation analogous to the fourth moment theorem for Gaussian limits?
• RQ4 How do the convergence properties of homogeneous sums behave in the context of Variance-Gamma approximation?
• RQ5 What is the role of moment convergence and cross-covariance decay in the multivariate convergence of chaotic vectors to a Variance-Gamma vector?

Key findings

• A sequence of random variables in the second Wiener chaos converges in distribution to a Variance-Gamma distribution if and only if their moments of order two through six converge to the corresponding moments of the target Variance-Gamma distribution.
• The paper establishes a quantitative bound on the total variation distance between a chaotic random variable and a Variance-Gamma distribution using Malliavin operators and contractions.
• For sequences converging to a Variance-Gamma distribution, the rate of convergence is governed by the maximum of the absolute values of the third and fourth cumulants, with explicit constants depending on the distribution parameters.
• Simplified bounds are derived for special cases such as the Laplace and symmetrized Gamma distributions.
• A multivariate extension is provided, showing that componentwise convergence and vanishing cross-covariances of squared components imply joint convergence to a multivariate Variance-Gamma distribution.
• A universality result is established for homogeneous sums, showing that convergence to a Variance-Gamma distribution is distribution-free under moment conditions.