Abstract
The variance-gamma (VG) distributions form a four-parameter family that includes as special and limiting cases the normal, gamma and Laplace distributions. Some of the numerous applications include financial modelling and approximation on Wiener space. Recently, Stein’s method has been extended to the VG distribution. However, technical difficulties have meant that bounds for distributional approximations have only been given for smooth test functions (typically requiring at least two derivatives for the test function). In this paper, which deals with symmetric variance-gamma (SVG) distributions, and a companion paper (Gaunt 2018), which deals with the whole family of VG distributions, we address this issue. In this paper, we obtain new bounds for the derivatives of the solution of the SVG Stein equation, which allow for approximations to be made in the Kolmogorov and Wasserstein metrics, and also introduce a distributional transformation that is natural in the context of SVG approximation. We apply this theory to obtain Wasserstein or Kolmogorov error bounds for SVG approximation in four settings: comparison of VG and SVG distributions, SVG approximation of functionals of isonormal Gaussian processes, SVG approximation of a statistic for binary sequence comparison, and Laplace approximation of a random sum of independent mean zero random variables.
Highlights
1.1 Overview of Stein’s Method for Variance-Gamma ApproximationThe variance-gamma (VG) distribution with parameters r > 0, θ ∈ R, σ > 0, μ ∈ R has probability density function p(x) = σ √π1 ( r 2 ) e θ σ2 (x −μ)√|x − μ| 2 θ2 + σ2 r −1 2 K r−1
The bounds obtained in [9] have a dependence on the test function h which means that error bounds for VG approximation can only be given for smooth test functions
In this paper and its companion [23], we obtain new bounds for the solution of the VG Stein equation that allow for Wasserstein and Kolmogorov error bounds for VG approximation via Stein’s method
Summary
Further VG approximations are given in [1] and [2], in which the limiting distribution is the difference of two centred gamma random variables. Introduced in 1972, Stein’s method [50] is a powerful tool for deriving distributional approximations with respect to a probability metric. There is active research into the development of Stein’s method for other distributional limits (see [30] for an overview), and Stein’s method for exponential and geometric approximation, for example, is well developed; see the survey [48]. The problem of bounding suph∈H |Eh(W ) − Eh(Z )| is reduced to bounding the solution (1.3) and some of its lower order derivatives and bounding the expectation on the right-hand side of (1.4). The bounds obtained in [9] have a dependence on the test function h which means that error bounds for VG approximation can only be given for smooth test functions
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