Stein’s method and exact Berry–Esseen asymptotics for functionals of Gaussian fields

By Ivan Nourdin, Giovanni Peccati. 2008

101 citations recorded by the source corpus.

We show how to detect optimal Berry–Esseen bounds in the normal approximation of functionals of Gaussian fields. Our techniques are based on a combination of Malliavin calculus, Stein’s method and the method of moments and cumulants, and provide de facto local (one-term) Edgeworth expansions. The findings of the present paper represent a further refinement of the main results proven in Nourdin and Peccati [Probab. Theory Related Fields 145 (2009) 75–118]. Among several examples, we discuss three crucial applications: (i) to Toeplitz quadratic functionals of continuous-time stationary processes (extending results by Ginovyan [Probab. Theory Related Fields 100 (1994) 395–406] and Ginovyan and Sahakyan [Probab. Theory Related Fields 138 (2007) 551–579]); (ii) to “exploding” quadratic functionals of a Brownian sheet; and (iii) to a continuous-time version of the Breuer–Major CLT for functionals of a fractional Brownian motion.

Subfields: Finance, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/473704bce4dd52a8bb620ca69a89c5a7777a013c

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Citations in the corpus

Multivariate normal approximation using Stein's method and Malliavin calculus (2008, methodology): “One crucial relation proved in [12], and then further exploited in [13], is the upper bound dW(Y,Z) ≤ E|1 − 〈DY,−DLY 〉H|.”
Quantitative Breuer-Major Theorems (2010, background): “We finally remark that the rate n for a = 2H− 2 ∈ (−2,−1] has been proved to be optimal in [27].”
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.”
The optimal fourth moment theorem (2013, unknown): “Our paper [13] provides some partial characterisation of optimal rates of convergence in the case where d = dKol (the Kolmogorov distance), whereas [4] contains similar findings for multidimensional CLTs and distances based on smooth mappings.”
Stein's method meets Malliavin calculus: a short survey with new estimates (2009, methodology): “- Our paper [19] is again based on Stein’s method and Malliavin calculus, and deals with the problem of determining optimal rates of convergence.”
Optimal Berry-Esseen rates on the Wiener space: the barrier of third and fourth cumulants (2011, unknown): “However, although quite flexible and far-reaching, the findings of [10] do not allow to deduce a complete solution (that is, a solution valid for arbitrary sequences {Fn} in a fixed Wiener chaos) of Problem 1.”
Exact maximum likelihood estimators for drift fractional Brownian motions (2009, methodology): “To obtain the Berry-Esséen bounds, we use the Stein’s method, following the work of [8].”
Cumulants on the Wiener Space (2009, methodology): “The techniques introduced in [13] have led to several applications and generalizations, for instance: in [14] one can find applications to Edgeworth expansions and reversed Berry-Esseen inequalities; [18] contains results for multivariate normal approximations; [1] focuses on further developments in”