2021/06/30 by Ehsan Azmoodeh, Azmoodeh, Ehsan, Peter Eichelsbacher +3 · 1 citation
Mathematics · #FOS: Mathematics #Geometry and complex manifolds #Probability (math.PR) #Random Matrices and Applications #Stochastic processes and statistical mechanics
paper · pdf · doi:10.48550/arxiv.2106.16018
openalex publication_date 2021/06/30 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
In this paper, we consider a target random variable Y ∼ \CVG 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 of difference of the first six cumulants. This six moment theorem 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.