Azmoodeh, Ehsan 
ORCID: 0000-0002-0401-793X, Eichelsbacher, Peter and Thaele, Christoph
(2022)
Optimal Variance-Gamma approximation on the second Wiener chaos.
JOURNAL OF FUNCTIONAL ANALYSIS, 282 (11).
p. 109450.
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Abstract
In this paper, we consider a target random variable $Y \sim \CVG$ distributed according to a centered Variance--Gamma distribution. For a generic random element $F=I_2(f)$ in the second Wiener chaos with $\E[F^2]= \E[Y^2]$ 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.
| Item Type: | Article |
|---|---|
| Additional Information: | 25 pages |
| Uncontrolled Keywords: | Six moment theorem, Stein's method, Variance-Gamma approximation, Wiener chaos |
| Divisions: | Faculty of Science and Engineering > School of Physical Sciences |
| Depositing User: | Symplectic Admin |
| Date Deposited: | 05 Jul 2021 12:47 |
| Last Modified: | 13 Mar 2023 22:57 |
| DOI: | 10.1016/j.jfa.2022.109450 |
| Open Access URL: | https://arxiv.org/pdf/2106.16018v1.pdf |
| Related URLs: | |
| URI: | https://livrepository.liverpool.ac.uk/id/eprint/3128873 |
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