Central limit theorems for multiple stochastic integrals and Malliavin calculus

By David Nualart, Salvador Ortiz. 2007

221 citations recorded by the source corpus.

We give a new characterization for the convergence in distribution to a standard normal law of a sequence of multiple stochastic integrals of a fixed order with variance one, in terms of the Malliavin derivatives of the sequence. We also give a new proof of the main theorem in [D. Nualart, G. Peccati, Central limit theorems for sequences of multiple stochastic integrals, Ann. Probab. 33 (2005) 177-193] using techniques of Malliavin calculus. Finally, we extend our result to the multidimensional case and prove a weak convergence result for a sequence of square integrable random vectors, giving an application.

Subfields: Finance, Management Science and Operations Research

View Paper: https://www.semanticscholar.org/paper/f337332afb1e28f4aa4494d64e7470eb521b5901

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Citations in the corpus

Chaos of a Markov operator and the fourth moment condition (2012, methodology): “In particular, the work by Nualart and Ortiz-Latorre [19] introduces a technological breakthrough with a new proof only based on Malliavin calculus and the use of integration by parts on Wiener space.”
Variations and Hurst index estimation for a Rosenblatt process using longer filters (2009, methodology): “/Rosenblatt variations using longer filters 1397 for multiple stochastic integrals based on the Malliavin calculus (see [22, 23]), we will see that the answer to the first question is negative, while the answer to the second question is affirmative.”
Parameter estimation for a partially observed Ornstein–Uhlenbeck process with long-memory noise (2015, methodology): “Finally, we will use the following central limit theorem for multiple stochastic integrals (see [24]).”
Absolute continuity and convergence of densities for random vectors on Wiener chaos (2012, background): “As a consequence of the main result in Nualart and Ortiz-Latorre [14], we deduce that Γn → C in L(2)(Ω) as n → ∞.”
Noncentral convergence of multiple integrals (2007, methodology): “We start by observing that, due to the multiplication formulae between stochastic integrals (see Proposition 1.1.3 in Nualart [9]), we have and (see also Nualart and Ortiz-Latorre [10, Lemma 2]).”
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.”
Central Limit Theorems for Multiple Skorokhod Integrals (2010, methodology): “In this case, Nualart and Peccati criterion combined with Lemma 6 in [21] tells us that, if the sequence of variances converges to one, then condition (a) is automatically satisfied.”
Stein’s method and exact Berry–Esseen asymptotics for functionals of Gaussian fields (2008, unknown): “, in terms of contraction operators); (ii) ( ) -> 0 as -> oo; and (iii) ( ) is directly related to quantities playing a fundamental role in the CLTs for functionals of Gaussian fields proven in [22, 23, 25, 26].”
Exact maximum likelihood estimators for drift fractional Brownian motions (2009, methodology): “ 1/2 is similar. We deﬁne GN = r N 2 (ˆσ2 −σ2) = σ2 √ 2N [(BH t) ′Γ−1 H B H − (t′Γ−1 H B H t ) 2 ttΓ−1 H t ]− r N 2 σ2 . From (2.7), it is obvious that E[G2 N] converges to σ4. Thus from Theorem 4 of [8] to show (3.1), it sufﬁces to show that k DGN k2 H −−−−→L2(Ω) C. First, using the deﬁnition of Ma”
Bipower Variation for Gaussian Processes with Stationary Increments (2009, unknown): “) The techniques used there, as well as in the present paper which considers the bipower case, come from very powerful recent results developed in the context of Wiener/Itô/Malliavin calculus, especially by Nualart, Peccati, and coauthors; see [24], [25], and [26] (cf.”
Invariance Principles for Homogeneous Sums: Universality of Gaussian Wiener Chaos (2009, methodology): “3 (See [28, 29]) Fix d > 2, and let Fn = Id(hn), n > 1 be a sequence in the dth Wiener chaos of G.”
Multi-Dimensional Gaussian Fluctuations on the Poisson Space (2010, unknown): “[9, 11, 13, 14, 18, 20].”
Parameter estimation for fractional Ornstein–Uhlenbeck processes of general Hurst parameter (2017, methodology): “The multidimensional version of the above theorem is also stated and proved in [17, 20, 21].”
Stein's method meets Malliavin calculus: a short survey with new estimates (2009, methodology): “This also refines the CLTs on Wiener chaos proved in [26] and [27].”
Parameter estimation for fractional Ornstein–Uhlenbeck processes (2009, methodology): “The proof of the central limit theorem is based on the characterization of the convergence in law for multiple stochastic integrals using the techniques of Malliavin calculus, established recently by Nualart and Ortiz-Latorre in [12].”
Stein’s method on Wiener chaos (2007, methodology): “3, where we collect the main findings of [36,37].”