Gaussian Limits for Vector-valued Multiple Stochastic Integrals

By Giovanni Peccati, Ciprian A. Tudor. 2005

270 citations recorded by the source corpus.

We establish necessary and sufficient conditions for a sequence of d-dimensional vectors of multiple stochastic integrals \(\mathbf{F}_{d}^{k} = (F_{1}^{k}, \dots, F_{d}^{k})\), \(k\geq 1\), to converge in distribution to a d-dimensional Gaussian vector \(\mathbf{N}_{d} = (N_{1}, \dots, N_{d}) \). In particular, we show that if the covariance structure of F d k converges to that of N d , then componentwise convergence implies joint convergence. These results extend to the multidimensional case the main theorem of [10].

Subfields: Applied Mathematics, Finance, Statistics and Probability

View Paper: https://www.semanticscholar.org/paper/7fb9172e64337f45b93b23dac3c972ca89f247cc

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Citations in the corpus

Convergence in total variation on Wiener chaos (2012, background): “5 The Peccati-Tudor theorem holds in total variation Let us rst recall the Peccati-Tudor theorem [22].”
Second order Poincaré inequalities and CLTs on Wiener space (2008, result): “This result generalizes and unifies the findings contained in [16, 20, 21, 23], and virtually closes a very fruitful circle of recent ideas linking Malliavin calculus, Stein’s method and central limit theorems (CLTs) on Wiener space (see also [15]).”
Stein’s method on Wiener chaos (2007, methodology): “One should compare this finding with the CLTs proved in [39], as well as the Gaussian approximations established in [38].”
Limit Theorems for Functionals of Higher Order Differences of Brownian Semi-Stationary Processes (2009, unknown): “which has been originally introduced in [19], [20] and [21], to show the consistency, i.”
Invariance Principles for Homogeneous Sums: Universality of Gaussian Wiener Chaos (2009, methodology): “For the proof of Theorem 7.5, we need the following result, which consists in a collection of some of the findings contained in the papers by Peccati and Tudor [34].”
Multipower Variation for Brownian Semistationary Processes (2011, methodology): “The basic tool we establish and apply for this is a normal central limit theorem for triangular arrays of dependent Gaussian variables.”
Stein's method meets Malliavin calculus: a short survey with new estimates (2009, unknown): “[39℄ C.A. Tudor and F. Viens (2008).”
Power variation of some integral fractional processes (2006, methodology): “To prove these results we make use of a central limit theorem for multiple stochastic integrals proved in Nualart and Peccati (2005), Peccati and Tudor (2005) and Hu and Nualart (2005).”
Parameter estimation for fractional Ornstein–Uhlenbeck processes of general Hurst parameter (2017, background): “Remark 2 The multidimensional version of the above theorem is also stated and proved in Nualart and Ortiz-Latorre (2008), Nourdin and Peccati (2012), Peccati and Tudor (2004).”
Rate of convergence and asymptotic error distribution of Euler approximation schemes for fractional diffusions (2014, background): “According to Peccati and Tudor [27] (see also Theorem 6.”
Stein’s method and exact Berry–Esseen asymptotics for functionals of Gaussian fields (2008, unknown): “04 58 v3 [ m at h. PR ] 9 D ec 2 00 9 The Annals of Probability 2009, Vol. 37, No. 6, 2231–2261 DOI: 10.1214/09-AOP461 c© Institute of Mathematical Statistics, 2009”
A central limit theorem for the Euler characteristic of a Gaussian excursion set (2016, methodology): “Our second tool is the Stein method, recently revisited in [25, 27, 28] for instance.”
Absolute continuity and convergence of densities for random vectors on Wiener chaos (2012, background): “Our first application of Proposition 5.1 consists in strengthening the celebrated PeccatiTudor [15] criterion of asymptotic normality.”
Multivariate normal approximation using Stein's method and Malliavin calculus (2008, methodology): “Notre travail généralise et affine des résultats antérieurs prouvés par Peccati et Tudor (2005), Nualart et Ortiz-Latorre (2007), Peccati (2007) et Nourdin et Peccati (2007b, 2008).”
Cumulants on the Wiener Space (2009, unknown): “The starting point of this active line of research is the paper [21], where the authors use Malliavin calculus in order to reﬁne some criteria for asymptotic normality on a ﬁxed Wiener chaos, originally proved in [22, 24] (see also [12] for some non-central version of these results).”