Electronic Journal of Probability

Malliavin-Stein method for variance-gamma approximation on Wiener space

Peter Eichelsbacher and Christoph Thäle

Full-text: Open access

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.

Article information

Source
Electron. J. Probab. Volume 20 (2015), 28 pp.

Dates
Accepted: 19 November 2015
First available in Project Euclid: 4 June 2016

Permanent link to this document
https://projecteuclid.org/euclid.ejp/1465067229

Digital Object Identifier
doi:10.1214/EJP.v20-4136

Mathematical Reviews number (MathSciNet)
MR3425543

Zentralblatt MATH identifier
1328.60065

Subjects
Primary: 60F05: Central limit and other weak theorems 60G15: Gaussian processes
Secondary: 60H05: Stochastic integrals 60H07: Stochastic calculus of variations and the Malliavin calculus

Keywords
Contractions cumulants Gaussian processes Laplace distribution Malliavin calculus non-central limit theorem rates of convergence Stein's method universality Variance-Gamma distribution Wiener chaos

Rights
This work is licensed under a Creative Commons Attribution 3.0 License.

Citation

Eichelsbacher, Peter; Thäle, Christoph. Malliavin-Stein method for variance-gamma approximation on Wiener space. Electron. J. Probab. 20 (2015), 28 pp. doi:10.1214/EJP.v20-4136. https://projecteuclid.org/euclid.ejp/1465067229.


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