Weak convergence on Wiener space: targeting the first two chaoses
Abstract
We consider sequences of random variables living in a finite sum of Wiener chaoses. We find necessary and sufficient conditions for convergence in law to a target variable living in the sum of the first two Wiener chaoses. Our conditions hold notably for sequences of multiple Wiener integrals. Malliavin calculus and in particular the Gamma-operators are used. Our results extend previous findings by Azmoodeh, Peccati and Poly (2014) and are applied to central and non-central convergence situations. Our methods are applied as well to investigate stable convergence. We finally exclude certain classes of random variables as target variables for sequences living in a fixed Wiener chaos.
Keywords
Cite
@article{arxiv.1701.06766,
title = {Weak convergence on Wiener space: targeting the first two chaoses},
author = {Christian Krein},
journal= {arXiv preprint arXiv:1701.06766},
year = {2019}
}
Comments
Minor typos have been corrected, Lemma 3.3 has been generalized, some new references have been added to the Introduction, the section about negative results has been summarized in Remark 4.10. This version corresponds to the paper published in ALEA