Fourth-Moment Theorems for Sums of Multiple Integrals
Fuente:
arXiv
Gespeichert in:
| Hauptverfasser: | , , |
|---|---|
| Format: | Preprint |
| Veröffentlicht: |
2025
|
| Schlagworte: | |
| Online-Zugang: | |
| Tags: |
Tag hinzufügen
Keine Tags, Fügen Sie den ersten Tag hinzu!
|
| _version_ | 1866912561188831232 |
|---|---|
| author | Basse-O'Connor, Andreas Kramer-Bang, David Svendsen, Clement |
| author_facet | Basse-O'Connor, Andreas Kramer-Bang, David Svendsen, Clement |
| contents | Nualart & Pecatti ([Nualart and Peccati, 2005, Thm 1]) established the first fourth-moment theorem for random variables in a fixed Wiener chaos, i.e. they showed that convergence of the sequence of fourth moments to the fourth moment of the standard Gaussian distribution is sufficient for weak convergence to the standard Gaussian. In this paper, we provide what we believe to be the first generalization to chaos expansions with more than a single term. Specifically, we show that a fourth-moment theorem holds for random variables consisting of sums of two multiple integrals of orders $p, q \in N$, where $p, q$ have different parities. Furthermore, we show that such random variables cannot themselves be Gaussian, again generalizing what is known for the fixed Wiener chaos setting. Finally, we show a fourth-moment theorem for variables with infinite Wiener chaos expansions when the terms in the expansions are independent and satisfy an additional regularity condition in terms of the Ornstein-Uhlenbeck operator. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2502_03596 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Fourth-Moment Theorems for Sums of Multiple Integrals Basse-O'Connor, Andreas Kramer-Bang, David Svendsen, Clement Probability 60F05, 60H07, 33C45 Nualart & Pecatti ([Nualart and Peccati, 2005, Thm 1]) established the first fourth-moment theorem for random variables in a fixed Wiener chaos, i.e. they showed that convergence of the sequence of fourth moments to the fourth moment of the standard Gaussian distribution is sufficient for weak convergence to the standard Gaussian. In this paper, we provide what we believe to be the first generalization to chaos expansions with more than a single term. Specifically, we show that a fourth-moment theorem holds for random variables consisting of sums of two multiple integrals of orders $p, q \in N$, where $p, q$ have different parities. Furthermore, we show that such random variables cannot themselves be Gaussian, again generalizing what is known for the fixed Wiener chaos setting. Finally, we show a fourth-moment theorem for variables with infinite Wiener chaos expansions when the terms in the expansions are independent and satisfy an additional regularity condition in terms of the Ornstein-Uhlenbeck operator. |
| title | Fourth-Moment Theorems for Sums of Multiple Integrals |
| topic | Probability 60F05, 60H07, 33C45 |
| url | https://arxiv.org/abs/2502.03596 |