Abstract
We establish explicit bounds on the convex distance between the distribution of a vector of smooth functionals of a Gaussian field and that of a normal vector with a positive-definite covariance matrix. Our bounds are commensurate to the ones obtained by Nourdin et al. (Ann Inst Henri Poincaré Probab Stat 46(1):45–58, 2010) for the (smoother) 1-Wasserstein distance, and do not involve any additional logarithmic factor. One of the main tools exploited in our work is a recursive estimate on the convex distance recently obtained by Schulte and Yukich (Electron J Probab 24(130):1–42, 2019). We illustrate our abstract results in two different situations: (i) we prove a quantitative multivariate fourth moment theorem for vectors of multiple Wiener–Itô integrals, and (ii) we characterize the rate of convergence for the finite-dimensional distributions in the functional Breuer–Major theorem.
| Original language | English |
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| Pages (from-to) | 2020-2037 |
| Number of pages | 18 |
| Journal | Journal of Theoretical Probability |
| Volume | 35 |
| Early online date | 4 Jun 2021 |
| DOIs | |
| Publication status | Published - 30 Sept 2022 |
Bibliographical note
Funding Information:We thank Simon Campese and Nicola Turchi for pointing out an error in an earlier version. I. Nourdin was supported by the FNR grant APOGee (R-AGR-3585-10) at Luxembourg University; G. Peccati is supported by the FNR grant FoRGES (R-AGR-3376-10) at Luxembourg University; X. Yang was supported by the FNR Grant MISSILe (R-AGR-3410-12-Z) at Luxembourg and Singapore Universities.
Keywords
- Breuer–Major Theorem
- Convex distance
- Fourth moment theorems
- Gaussian fields
- Malliavin–Stein method
- Multidimensional normal approximations
ASJC Scopus subject areas
- Statistics and Probability
- General Mathematics
- Statistics, Probability and Uncertainty