Wasserstein and Kolmogorov error bounds for variance-gamma approximation via Stein's method I

TL;DR

This paper develops new bounds for symmetric variance-gamma (SVG) distribution approximation using Stein's method, enabling error estimation in Wasserstein and Kolmogorov metrics across various applications.

Contribution

It introduces bounds for derivatives of the SVG Stein equation solutions and a natural distributional transformation for SVG approximation, extending Stein's method to less smooth test functions.

Findings

Derived bounds for SVG Stein equation solutions.

Achieved Wasserstein and Kolmogorov error bounds for SVG approximation.

Applied bounds to multiple practical approximation scenarios.

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 \cite{gaunt vgii}, 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.