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
∑i,j,k=1m,n,rXikYjk, 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.Comment: 39 pages. Published Versio