Optimal Variance–Gamma approximation on the second Wiener chaos

Abstract

In this paper, we consider a target random variable Y∼VGc(r,θ,σ) distributed according to a centered Variance–Gamma distribution. For a generic random element F=I2(f) in the second Wiener chaos with E[F2]=E[Y2] we establish a non-asymptotic optimal bound on the distance between F and Y in terms of the maximum difference of the first six cumulants. This six moment theorem complements and in some sense extends the celebrated optimal fourth moment theorem of I. Nourdin & G. Peccati for normal approximation. The main body of our analysis constitutes a splitting technique for test functions in the Banach space of Lipschitz functions relying on the compactness of the Stein operator. The recent developments around Stein method for Variance–Gamma approximation by R. Gaunt play a significant role in our study. As an application we consider the generalized Rosenblatt process at the extreme critical exponent, first studied by S. Bai & M. Taqqu.