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 $\sum_{i,j,k=1}^{m,n,r}X_{ik}Y_{jk}$, 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.
Highlights
In 1972, Stein [41] introduced a powerful method for deriving bounds for normal approximation
Stein’s method for normal approximation rests on the following characterization of the normal distribution, which can be found in Stein [42], namely Z ∼ N (μ, σ2) if and only if
The following proposition, which can be found in Bibby and Sørensen [7], shows that the class of Variance-Gamma distributions is closed under convolution, provided that the random variables have common values of θ and σ
Summary
In 1972, Stein [41] introduced a powerful method for deriving bounds for normal approximation. The class of Variance-Gamma distributions includes the Laplace distribution as a special case and in the appropriate limits reduces to the normal and Gamma distributions This family of distributions contains many other distributions that are of interest, which we list in the following proposition (the proof is given in Appendix A). One of the main results of this paper (see Lemma 3.1) is the following Stein equation for the Variance-Gamma distributions: σ2(x − μ)f (x) + (σ2r + 2θ(x − μ))f (x) + (rθ − (x − μ))f (x) = h(x) − VGrσ,,θμh, (1.7). We apply this bound to an application of binary sequence comparison, which is a simple special case of the more general problem of word sequence comparison. Appendix B provides a list of some elementary properties of modified Bessel functions that we make use of in this paper
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