Strong asymptotic independence on Wiener chaos

Nourdin, Ivan; Nualart, David; Peccati, Giovanni

doi:10.1090/proc12769

Cited by 16 publications

(24 citation statements)

References 15 publications

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“…Without a rate of convergence, Proposition 5.18 is also a consequence of the strong asymptotic independence properties inside the Wiener chaos. In particular, the result is a consequence of Theorem 1.4 in [10] and the fact that the distribution of each Y j , j = 1, . .…”

Section: The Case Of the Second Wiener Chaosmentioning

confidence: 88%

Malliavin-Stein method for variance-gamma approximation on Wiener space

Eichelsbacher

Thäle

2015

Electron. J. Probab.

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We combine Malliavin calculus with Stein's method to derive bounds for the Variance-Gamma approximation of functionals of isonormal Gaussian processes, in particular of random variables living inside a fixed Wiener chaos induced by such a process. The bounds are presented in terms of Malliavin operators and norms of contractions. We show that a sequence of distributions of random variables in the second Wiener chaos converges to a Variance-Gamma distribution if and only if their moments of order two to six converge to that of a Variance-Gamma distributed random variable (six moment theorem). Moreover, simplified versions for Laplace or symmetrized Gamma distributions are presented. Also multivariate extensions and a universality result for homogeneous sums are considered.

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Section: The Case Of the Second Wiener Chaosmentioning

confidence: 88%

Malliavin-Stein method for variance-gamma approximation on Wiener space

Eichelsbacher

Thäle

2015

Electron. J. Probab.

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“…We also take the liberty to point out an intermediate result on the joint convergence of stochastic processes in finite L 2 chaos, for it maybe of service. The proof is a slight modification of results in [29,41].…”

Section: Remark 13mentioning

confidence: 80%

“…In [8], vector valued combinations of short-and long-rangedependent sums were studied, however, the limit of each component is assumed to be moment determinate (which can only happen when they are in the L 2 chaos expansion of order less or equal to 2). This is due to a restriction in the asymptotic independence result in [33], which was extended in [29].…”

Section: A(n)mentioning

confidence: 99%

Functional Limit Theorems for the Fractional Ornstein–Uhlenbeck Process

Gehringer

2020

J Theor Probab

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We prove a functional limit theorem for vector-valued functionals of the fractional Ornstein–Uhlenbeck process, providing the foundation for the fluctuation theory of slow/fast systems driven by both long- and short-range-dependent noise. The limit process has both Gaussian and non-Gaussian components. The theorem holds for any $$L^2$$ L 2 functions, whereas for functions with stronger integrability properties the convergence is shown to hold in the Hölder topology, the rough topology for processes in $$C^{\frac{1}{2}+}$$ C 1 2 + . This leads to a ‘rough creation’ / ‘rough homogenization’ theorem, by which we mean the weak convergence of a family of random smooth curves to a non-Markovian random process with non-differentiable sample paths. In particular, we obtain effective dynamics for the second-order problem and for the kinetic fractional Brownian motion model.

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“…The sequences are supposed to converge in law to a non-zero target variable X and we ask whether the sequence also converges stably. Our first result follows from [12,Theorem 1.3]. As before, we shall write ϕ Y for the characteristic function of a random variable Y . "…”

Section: Stable Convergencementioning

confidence: 98%

Weak convergence on Wiener space: targeting the first two chaoses

Krein¹

2019

ALEA

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We consider sequences of random variables living in a finite sum of Wiener chaoses. We find necessary and sufficient conditions for convergence in law to a target variable living in the sum of the first two Wiener chaoses. Our conditions hold notably for sequences of multiple Wiener integrals. Malliavin calculus and in particular the Γ-operators are used. Our results extend previous findings by Azmoodeh, Peccati and Poly (2014) and are applied to central and non-central convergence situations. Our methods are applied as well to investigate stable convergence. We finally exclude certain classes of random variables as target variables for sequences living in a fixed Wiener chaos.MSC 2010 subject classifications: 60F05; 60G15; 60H07.

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Strong asymptotic independence on Wiener chaos

Cited by 16 publications

References 15 publications

Malliavin-Stein method for variance-gamma approximation on Wiener space

Malliavin-Stein method for variance-gamma approximation on Wiener space

Functional Limit Theorems for the Fractional Ornstein–Uhlenbeck Process

Weak convergence on Wiener space: targeting the first two chaoses

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