Multivariate Normal Approximation on the Wiener Space: New Bounds in the Convex Distance

Ivan Nourdin; Giovanni Peccati; Xiaochuan Yang

doi:10.1007/s10959-021-01112-6

Multivariate Normal Approximation on the Wiener Space: New Bounds in the Convex Distance

Ivan Nourdin, Giovanni Peccati, Xiaochuan Yang

Department of Mathematical Sciences

Université du Luxembourg

Research output: Contribution to journal › Article › peer-review

8 Citations (SciVal)

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
Pages (from-to)	2020-2037
Number of pages	18
Journal	Journal of Theoretical Probability
Volume	35
Early online date	4 Jun 2021
DOIs	https://doi.org/10.1007/s10959-021-01112-6
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.

Funding

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

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10.1007/s10959-021-01112-6Licence: CC BY

Cite this

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title = "Multivariate Normal Approximation on the Wiener Space: New Bounds in the Convex Distance",

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 Poincar{\'e} 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–It{\^o} integrals, and (ii) we characterize the rate of convergence for the finite-dimensional distributions in the functional Breuer–Major theorem.",

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author = "Ivan Nourdin and Giovanni Peccati and Xiaochuan Yang",

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. ",

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PY - 2022/9/30

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N2 - 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.

AB - 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.

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Multivariate Normal Approximation on the Wiener Space: New Bounds in the Convex Distance

Abstract

Bibliographical note

Funding

Keywords

ASJC Scopus subject areas

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