Central limit theorems for sequences of multiple stochastic integrals

By David Nualart, Giovanni Peccati. 2005

471 citations recorded by the source corpus.

We characterize the convergence in distribution to a standard normal law for a sequence of multiple stochastic integrals of a fixed order with variance converging to 1. Some applications are given, in particular to study the limiting behavior of quadratic functionals of Gaussian processes.

Subfields: Finance, Management Science and Operations Research

View Paper: https://www.semanticscholar.org/paper/0b759d3dedfa36dcdf5ed7f403c7978d3a3654f5

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Citations in the corpus

Parameter Estimation for Fractional Ornstein-Uhlenbeck Processes: Non-ergodic Case (2011, methodology): “[12] D. Nualart and G. Peccati.”
Variations and Hurst index estimation for a Rosenblatt process using longer filters (2009, methodology): “Our analysis will be based on the following result, due to Nualart and Peccati (see Theorem 1 in [22]).”
Stein's Method and Stochastic Analysis of Rademacher Functionals (2008, unknown): “Technical proofs are deferred to the Appendix.”
Optimal Berry-Esseen rates on the Wiener space: the barrier of third and fourth cumulants (2011, unknown): “One can also prove that, if F belongs to a fixed Wiener chaos and has unit variance, then E[F 4] > 3 (see [17]).”
Fourth Moment Theorems for Markov Diffusion Generators (2013, methodology): “Inspired by the insightful article [4], we revisit the Nualart-Peccati-criterion [13] (now known as the Fourth Moment Theorem) from the point of view of spectral theory of general Markov diUusion generators.”
Convergence in total variation on Wiener chaos (2012, unknown): “Keywords: Convergence in distribution; Convergence in total variation; Malliavin calculus; multiple Wiener-Itô integral; Wiener chaos.”
Multivariate normal approximation using Stein's method and Malliavin calculus (2008, background): “4) yields bounds that are intimately related with the CLTs proved in [15] and [16].”
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].”
Noncentral convergence of multiple integrals (2007, background): “More recently, Nualart and Peccati [11] proved the following result (here, and for the rest of the paper, we shall denote by N (0 , 1) the law of a Gaussian random variable with zero mean and unit variance).”
Asymptotics for spherical needlets (2006, background): “See also Nualart and Peccati (2005) for a more recent point of view.”
The Variance-Gamma Distribution: A Review (2023, unknown): “For f ∈ H 2, we denote the double Wiener-Itô integral by I2(f) (see Nourdin and Peccati [88, Section 2.7] for a definition and fundamental properties).”
Gaussian Limits for Vector-valued Multiple Stochastic Integrals (2005, methodology): “Now, since (4) implies E [( ni!J 1 ni ( f (k) i ))2] → k↑+∞ 1 for every i, condition (i) yields – thanks to Proposition 3 in [9] – that”
The optimal fourth moment theorem (2013, background): “1 (Fourth moment theorem [12, 20]) Fix an integer q > 2 and let N ∼ N (0, 1) be a standard Gaussian random variable.”
Stein’s method on Wiener chaos (2007, methodology): “3, where we collect the main findings of [36,37].”
New Berry-Esseen bounds for non-linear functionals of Poisson random measures (2013, background): “3 should be compared with the following result from [20]: Let for some integer q ≥ 2, IG q (fn) be a sequence of multiple integrals with respect to a Gaussian random measure on Z such that for each, n ∈ N, fn ∈ L2(μq) is symmetric (but not necessarily non-negative).”
Invariance Principles for Homogeneous Sums: Universality of Gaussian Wiener Chaos (2009, background): “, [20]) that, if F is a nonzero element of the dth Wiener chaos of a given Gaussian sequence (d ≥ 2), then κ4(F ) > 0.”
Cumulants on the Wiener Space (2009, background): “22)) of the following result, first proved in [22] and yielding a necessary and sufficient condition for CLTs on a fixed chaos.”
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.”