Stein factors for variance-gamma approximation in the Wasserstein and Kolmogorov distances

Abstract

We obtain new bounds for the solution of the variance-gamma (VG) Stein equation that are of the correct form for approximations in terms of the Wasserstein and Kolmorogorov metrics. These bounds hold for all parameters values of the four parameter VG class. As an application we obtain explicit Wasserstein and Kolmogorov distance error bounds in a six moment theorem for VG approximation of double Wiener-Itô integrals.

Gamma, Gaussian and Poisson approximations for random sums using size-biased and generalized zero-biased couplings

“…We also note that the gamma distribution is a limiting case of the variance-gamma distribution, for which Stein's method was first developed by Gaunt [6]. For more recent work in this area, see [7] and references therein.…”

Bounding Kolmogorov distances through Wasserstein and related integral probability metrics

“…(3.21) (iii): Finally, we suppose 0 < r < 1. Part (i) of Lemma 5.1 of [26] tells us that e x x µ K µ (x) ≤ 2 µ−1 Γ(µ), for x > 0, 0 < µ ≤ 1 2 . But also, K ν (x) = K −ν (x) for all x ∈ R (see [50]), and so…”

“…Basic properties of the VG distribution are given in [23,36]. Upper bounds on d K (X, Y ) for an arbitrary random variable X and Y ∼ VG(r, θ, σ, µ) in terms of d [1] (X, Y ) = d W (X, Y ) were given by [25] for the case θ = 0 and by [26] for general θ ∈ R. We now apply Proposition 2.1 to provide an upper bound on…”

“…For the case Y has a variance-gamma distribution, [25,26] obtained analogues of (1.2) in which the variance-gamma density has a singularity at the location parameter, whilst [43] have obtained analogues of (1.2) for the case that Y is a mixture of normal distributions. Multivariate generalisations of (1.2) in which Y has the multivariate normal distribution are given in [5,33] with a similar bound on the convex distance recently obtained by [48].…”

Bounds for an integral involving the modified Struve function of the first kind

“…For 0 < β < 1 there does not exist a simple closed-form formula for this integral. The inequalities of [12,14,18] played a crucial role in the development of Stein's method [9,27,32] for variance-gamma approximation [10,11,17,19]. As the inequalities of [12,14,18] are simple and surprisingly accurate, they may also be useful in other problems involving modified Bessel functions; see, for example, [7,8] in which inequalities for modified Bessel functions of the first kind were used to derive tight bounds for the generalized Marcum Q-function, which arises in radar signal processing.…”