Variance-Gamma approximation via Stein's method

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.Comment: 39 pages. Published Versio

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

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.Comment: 37 pages, to appear in Journal of Theoretical Probability, 2018

Rates of convergence in normal approximation under moment conditions via new bounds on solutions of the Stein equation

New bounds for the $k$-th order derivatives of the solutions of the normal and multivariate normal Stein equations are obtained. Our general order bounds involve fewer derivatives of the test function than those in the existing literature. We apply these bounds and local approach couplings to obtain an order $n^{-(p-1)/2}$ bound, for smooth test functions, for the distance between the distribution of a standardised sum of independent and identically distributed random variables and the standard normal distribution when the first $p$ moments of these distributions agree. We also obtain a bound on the convergence rate of a sequence of distributions to the normal distribution when the moment sequence converges to normal moments.Comment: 16 pages. Final version. To appear in Journal of Theoretical Probabilit

Derivative formulas for Bessel, Struve and Anger-Weber functions

We derive formulas for the derivatives of general order for the functions $z^{-\nu}h_{\nu}(z)$ and $z^{\nu}h_{\nu}(z)$, where $h_{\nu}(z)$ is a Bessel, Struve or Anger--Weber function.Comment: 9 page

Inequalities for modified Bessel functions and their integrals

Simple inequalities for some integrals involving the modified Bessel functions $I_{\nu}(x)$ and $K_{\nu}(x)$ are established. We also obtain a monotonicity result for $K_{\nu}(x)$ and a new lower bound, that involves gamma functions, for $K_0(x)$.Comment: 13 pages. Final version. To appear in Journal of Mathematical Analysis and Application

Inequalities for integrals of modified Bessel functions and expressions involving them

Simple inequalities are established for some integrals involving the modified Bessel functions of the first and second kind. In most cases, we show that we obtain the best possible constant or that our bounds are tight in certain limits. We apply these inequalities to obtain uniform bounds for several expressions involving integrals of modified Bessel functions. Such expressions occur in Stein's method for variance-gamma approximation, and the results obtained in this paper allow for technical advances in the method. We also present some open problems that arise from this research.Comment: 20 pages. Final version. To appear in Journal of Mathematical Analysis and Application

Inequalities for integrals of the modified Struve function of the first kind

Simple inequalities for some integrals involving the modified Struve function of the first kind $\mathbf{L}_{\nu}(x)$ are established. In most cases, these inequalities have best possible constant. We also deduce a tight double inequality, involving the modified Struve function $\mathbf{L}_{\nu}(x)$, for a generalized hypergeometric function.Comment: 9 pages. To appear in Results in Mathematics, 2018