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2014 Variance-Gamma approximation via Stein's method
Robert Gaunt
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Electron. J. Probab. 19: 1-33 (2014). DOI: 10.1214/EJP.v19-3020


Variance-Gamma distributions are widely used in financial modelling and contain as special cases the normal, Gamma and Laplace distributions. In this paper we extend Stein's method to this class of distributions. In particular, we obtain a Stein equation and smoothness estimates for its solution. This Stein equation has the attractive property of reducing to the known normal and Gamma Stein equations for certain parameter values. We apply these results and local couplings to bound the distance between sums of the form $\sum_{i,j,k=1}^{m,n,r}X_{ik}Y_{jk}$, where the $X_{ik}$ and $Y_{jk}$ are independent and identically distributed random variables with zero mean, by their limiting Variance-Gamma distribution. Through the use of novel symmetry arguments, we obtain a bound on the distance that is of order $m^{-1}+n^{-1}$ for smooth test functions. We end with a simple application to binary sequence comparison.


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Robert Gaunt. "Variance-Gamma approximation via Stein's method." Electron. J. Probab. 19 1 - 33, 2014.


Accepted: 29 March 2014; Published: 2014
First available in Project Euclid: 4 June 2016

zbMATH: 1291.60046
MathSciNet: MR3194737
Digital Object Identifier: 10.1214/EJP.v19-3020

Primary: 60F05

Keywords: rates of convergence , Stein's method , Variance-Gamma approximation


Vol.19 • 2014
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