Wasserstein and Kolmogorov Error Bounds for Variance-Gamma Approximation via Stein’s Method I

Robert E. Gaunt*

*Corresponding author for this work

    Research output: Contribution to journalArticlepeer-review

    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.

    Original languageEnglish
    JournalJournal of Theoretical Probability
    Early online date1 Nov 2018
    DOIs
    Publication statusPublished - 2018

    Keywords

    • Distributional transformation
    • Rate of convergence
    • Stein’s method
    • Variance-gamma approximation

    Fingerprint

    Dive into the research topics of 'Wasserstein and Kolmogorov Error Bounds for Variance-Gamma Approximation via Stein’s Method I'. Together they form a unique fingerprint.

    Cite this