Stein’s method on Wiener chaos

By Ivan Nourdin, Giovanni Peccati. 2007

447 citations recorded by the source corpus.

We combine Malliavin calculus with Stein’s method, in order to derive explicit bounds in the Gaussian and Gamma approximations of random variables in a fixed Wiener chaos of a general Gaussian process. Our approach generalizes, refines and unifies the central and non-central limit theorems for multiple Wiener–Itô integrals recently proved (in several papers, from 2005 to 2007) by Nourdin, Nualart, Ortiz-Latorre, Peccati and Tudor. We apply our techniques to prove Berry–Esséen bounds in the Breuer–Major CLT for subordinated functionals of fractional Brownian motion. By using the well-known Mehler’s formula for Ornstein–Uhlenbeck semigroups, we also recover a technical result recently proved by Chatterjee, concerning the Gaussian approximation of functionals of finite-dimensional Gaussian vectors.

Subfields: Applied Mathematics, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/8c301d5216357628389b6c27a4ed189e9e8e77c6

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Citations in the corpus

Chi-square approximation by Stein's method with application to Pearson's statistic (2015, methodology): “rs, Stein’s method has been extended to many other distributions, such as the Poisson [5], multinomial [20], exponential [4], [27], Laplace [29], variance-gamma [14], and the gamma distribution [21], [25], which we develop further in this paper. For multivariate normal approximations, the Gaunt etal”
Stein’s method for invariant measures of diffusions via Malliavin calculus☆ (2012, unknown): “Then we will consider the generator of the diffusion X above and we will use the properties of this operator and the integration by parts on Wiener space in the spirit of [7] in order to obtain a bound between the law of an arbitrary random variable F and the law with density p .”
Variations and estimators for selfsimilarity parameters via Malliavin calculus (2009, background): “We refer to [19] for the one dimensional case and H ∈ (0, 3 4), [2] for then one-dimensional case and H ∈ [ 4 , 1) and to [20] for the multidimensional case and H ∈ (0, 34 ).”
Quantitative Breuer-Major Theorems (2010, background): “The next statement, which is the main result of the paper, asserts that the quantities deﬁned above can be used to write explicit bounds of the type (1.5).”
On Stein's Method for Infinitely Divisible Laws with Finite First Moment (2017, background): “several quantitative convergence results follow). (iii) As ﬁrst noticed in [41], the quantity 1 2 Xn k=1 Z+N −N k Kν(t,N) n −K(t,N) dt, is the L1 analog of the L2 Stein discrepancy considered e.g. in [7, 8, 26, 21]. As such, it seems to be the appropriate Stein quantity to deal with sums of independ”
High-dimensional central limit theorems by Stein’s method (2020, methodology): “1 is also interesting in the context of the so-called Malliavin–Stein method (see Nourdin and Peccati (2012) for an exposition of this topic).”
Error bounds on the non-normal approximation of Hermite power variations of fractional Brownian motion (2008, methodology): “The current paper, together with [ 11 ], seem to be the first attempt in such direction.”
Central Limit Theorems for Multiple Skorokhod Integrals (2010, methodology): “calculus. Finally, the Berry-Esseen’s type bound (1.2) is taken from Nourdin and Peccati [ 16 ], while (1.3) was shown in Nourdin, Peccati and Reinert [17].”
Stein couplings for normal approximation (2010, background): “A more recent development was initiated by Nourdin and Peccati (2009) and related articles, where a fruitful theory for normal approximation for functionals of Gaussian measures, Rademacher sequences and Poisson measures is developed.”