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

Ivan Nourdin, Giovanni Peccati, Xiaochuan Yang

Research output: Contribution to journalArticlepeer-review

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 languageEnglish
JournalJournal of Theoretical Probability
Early online date4 Jun 2021
DOIs
Publication statusE-pub ahead of print - 4 Jun 2021

Keywords

  • Breuer–Major Theorem
  • Convex distance
  • Fourth moment theorems
  • Gaussian fields
  • Malliavin–Stein method
  • Multidimensional normal approximations

ASJC Scopus subject areas

  • Statistics and Probability
  • Mathematics(all)
  • Statistics, Probability and Uncertainty

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