Multivariate normal approximation using Stein's method and Malliavin calculus

By Ivan Nourdin, Giovanni Peccati, Anthony R'eveillac. 2008

151 citations recorded by the source corpus.

We combine Stein's method with Malliavin calculus in order to obtain explicit bounds in the multidimensional normal approximation (in the Wasserstein distance) of functionals of Gaussian fields. Our results generalize and refine the main findings by Peccati and Tudor (2005), Nualart and Ortiz-Latorre (2007), Peccati (2007) and Nourdin and Peccati (2007b, 2008); in particular, they apply to approximations by means of Gaussian vectors with an arbitrary, positive definite covariance matrix. Among several examples, we provide an application to a functional version of the Breuer-Major CLT for fields subordinated to a fractional Brownian motion.

Subfields: Finance, Statistics and Probability

View Paper: https://www.semanticscholar.org/paper/108f3af9424510b6c545037a22e00d7f045fb43f

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Citations in the corpus

Multi-Dimensional Gaussian Fluctuations on the Poisson Space (2010, methodology): “These results are taken from: [8] (Line 1), [10] (Line 2), [17] (Line 3) and Theorem 3.”
Stein’s method and Normal approximation of Poisson functionals (2008, background): “See [15, 18] and [26] for similar results involving vectors of multiple integrals (of arbitrary order) with respect to Gaussian random measures.”
Stein's method meets Malliavin calculus: a short survey with new estimates (2009, background): “1 Main bounds We shall now present (without proof) a result taken from [23], concerning the Gaussian approximation of vectors of random variables that are differentiable in the Malliavin sense.”
Invariance Principles for Homogeneous Sums: Universality of Gaussian Wiener Chaos (2009, methodology): “The findings of [12] are extended in [13] and [17], dealing respectively with lower bounds and multidimensional normal approximations.”
Second order Poincaré inequalities and CLTs on Wiener space (2008, background): “The following statement, whose proof is based on the results obtained in [18], provides a multidimensional version of (1.”
Measuring Sample Quality with Diffusions (2016, background): “11 Example 1 (Bayesian logistic regression with Gaussian prior).”
Stein kernels and moment maps (2018, background): “It would also be possible to build a Stein kernel using the construction of [10, 31] and the optimal transport map sending the standard Gaussian measure onto μ.”
Variations and Hurst index estimation for a Rosenblatt process using longer filters (2009, background): “Recent interest in the Rosenblatt and other Hermite processes, due in part to their non-Gaussian character, and in part for their independent mathematical value, is evidenced by the following references: [4, 6, 10, 18, 19, 20, 27, 28].”
Stein's Method and Stochastic Analysis of Rademacher Functionals (2008, methodology): “The papers [22, 23, 24] deal with Stein’s method and Malliavin calculus (see e.g. [26]) in a Gaussian setting; in [25] one can ﬁnd extensions involving density estimates and concentration inequalities; the paper [28] is devoted to explicit bounds, obtained by combining Malliavin calculus and…”