Electronic Journal of Probability

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

Peter Eichelsbacher and Christoph Thäle

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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

Electron. J. Probab., Volume 20 (2015), paper no. 123, 28 pp.

Received: 23 February 2015
Accepted: 9 November 2015
First available in Project Euclid: 4 June 2016

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Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

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

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

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Eichelsbacher, Peter; Thäle, Christoph. Malliavin-Stein method for variance-gamma approximation on Wiener space. Electron. J. Probab. 20 (2015), paper no. 123, 28 pp. doi:10.1214/EJP.v20-4136. https://projecteuclid.org/euclid.ejp/1465067229

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