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2015 Malliavin-Stein method for variance-gamma approximation on Wiener space
Peter Eichelsbacher, Christoph Thäle
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Electron. J. Probab. 20: 1-28 (2015). DOI: 10.1214/EJP.v20-4136


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.


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Peter Eichelsbacher. Christoph Thäle. "Malliavin-Stein method for variance-gamma approximation on Wiener space." Electron. J. Probab. 20 1 - 28, 2015.


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

zbMATH: 1328.60065
MathSciNet: MR3425543
Digital Object Identifier: 10.1214/EJP.v20-4136

Primary: 60F05 , 60G15
Secondary: 60H05 , 60H07

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


Vol.20 • 2015
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