Variance-Gamma approximation via Stein's method

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    Abstract

    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 ∑m,n,ri,j,k=1XikYjk, where the Xik and Yjk 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.
    Original languageEnglish
    Article number38
    Number of pages33
    JournalElectronic Journal of Probability
    Volume19
    DOIs
    Publication statusPublished - 29 Mar 2014

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