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Stein’s method on Wiener chaos
 Probab. Theory Relat. Fields
, 2009
"... Abstract: We combine Malliavin calculus with Stein’s method, in order to derive explicit bounds in the Gaussian and Gamma approximations of random variables in a fixed Wiener chaos of a general Gaussian process. Our approach generalizes, refines and unifies the central and noncentral limit theorems ..."
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Cited by 51 (31 self)
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Abstract: We combine Malliavin calculus with Stein’s method, in order to derive explicit bounds in the Gaussian and Gamma approximations of random variables in a fixed Wiener chaos of a general Gaussian process. Our approach generalizes, refines and unifies the central and noncentral limit theorems for multiple WienerItô integrals recently proved (in several papers, from 2005 to 2007) by Nourdin, Nualart, OrtizLatorre, Peccati and Tudor. We apply our techniques to prove BerryEsséen bounds in the BreuerMajor CLT for subordinated functionals of fractional Brownian motion. By using the wellknown Mehler’s formula for OrnsteinUhlenbeck semigroups, we also recover a technical result recently proved by Chatterjee, concerning the Gaussian approximation of functionals of finitedimensional Gaussian vectors.
Central limit theorems for multiple Skorohod integrals
, 2008
"... In this paper, we prove a central limit theorem for a sequence of multiple Skorohod integrals using the techniques of Malliavin calculus. The convergence is stable, and the limit is a conditionally Gaussian random variable. Some applications to sequences of multiple stochastic integrals, and renorma ..."
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Cited by 34 (9 self)
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In this paper, we prove a central limit theorem for a sequence of multiple Skorohod integrals using the techniques of Malliavin calculus. The convergence is stable, and the limit is a conditionally Gaussian random variable. Some applications to sequences of multiple stochastic integrals, and renormalized weighted Hermite variations of the fractional Brownian motion are discussed. Key words: central limit theorem, fractional Brownian motion, Malliavin calculus.
Stein’s method meets malliavin calculus: a short survey with new estimates.
, 2009
"... Abstract: We provide an overview of some recent techniques involving the Malliavin calculus of variations and the socalled Stein's method for the Gaussian approximations of probability distributions. Special attention is devoted to establishing explicit connections with the classic method of ..."
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Cited by 24 (7 self)
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Abstract: We provide an overview of some recent techniques involving the Malliavin calculus of variations and the socalled Stein's method for the Gaussian approximations of probability distributions. Special attention is devoted to establishing explicit connections with the classic method of moments: in particular, we use interpolation techniques in order to deduce some new estimates for the moments of random variables belonging to a xed Wiener chaos. As an illustration, a class of central limit theorems associated with the quadratic variation of a fractional Brownian motion is studied in detail.
Estimators of fractal dimension: Assessing the roughness of time series and spatial data
, 2010
"... Lies, damn lies, and dimension estimates ..."
Multipower variation for Brownian semistationary processes (full version). CREATES research paper 200921, Aarhus Univ. Available at http://www.econ.au.dk/research/researchcentres/creates/research/researchpapers/ researchpapers2009/. Asymptotic result
, 2009
"... In this paper we study the asymptotic behaviour of power and multipower variations of processes Y : where g : (0, ∞) → R is deterministic, σ > 0 is a random process, W is the stochastic Wiener measure and Z is a stochastic process in the nature of a drift term. Processes of this type serve, in p ..."
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Cited by 11 (9 self)
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In this paper we study the asymptotic behaviour of power and multipower variations of processes Y : where g : (0, ∞) → R is deterministic, σ > 0 is a random process, W is the stochastic Wiener measure and Z is a stochastic process in the nature of a drift term. Processes of this type serve, in particular, to model data of velocity increments of a fluid in a turbulence regime with spot intermittency σ . The purpose of this paper is to determine the probabilistic limit behaviour of the (multi)power variations of Y as a basis for studying properties of the intermittency process σ . Notably the processes Y are in general not of the semimartingale kind and the established theory of multipower variation for semimartingales does not suffice for deriving the limit properties. As a key tool for the results, a general central limit theorem for triangular Gaussian schemes is formulated and proved. Examples and an application to the realised variance ratio are given.
Moments, cumulants and diagram formulae for nonlinear functionals of random measures
, 2008
"... This survey provides a unified discussion of multiple integrals, moments, cumulants and diagram formulae associated with functionals of completely random measures. Our approach is combinatorial, as it is based on the algebraic formalism of partition lattices and Möbius functions. Gaussian and Poisso ..."
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Cited by 10 (6 self)
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This survey provides a unified discussion of multiple integrals, moments, cumulants and diagram formulae associated with functionals of completely random measures. Our approach is combinatorial, as it is based on the algebraic formalism of partition lattices and Möbius functions. Gaussian and Poisson measures are treated in great detail. We also present several combinatorial interpretations of some recent CLTs involving sequences of random variables belonging to a fixed Wiener chaos.
Stein’s method, Malliavin calculus and infinitedimensional Gaussian analysis
 LECTURE NOTES, WORKSHOP ON STEIN’S METHOD, SINGAPORE
, 2009
"... This expository paper is a companion of the four onehour tutorial lectures given in the occasion of the special month Progress in Stein’s Method, held at the University of Singapore in January 2009. We will explain how one can combine Stein’s method with Malliavin calculus, in order to obtain expli ..."
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Cited by 3 (0 self)
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This expository paper is a companion of the four onehour tutorial lectures given in the occasion of the special month Progress in Stein’s Method, held at the University of Singapore in January 2009. We will explain how one can combine Stein’s method with Malliavin calculus, in order to obtain explicit bounds in the normal and Gamma approximation of functionals of infinitedimensional Gaussian fields. The core of our discussion is based on a series of papers jointly written with I. Nourdin, as well as with I. Nourdin and A. Réveillac.
A distributional limit theorem for the realized power variation of linear fractional stable motions/MathPreprints A DISTRIBUTIONAL LIMIT THEOREM FOR THE REALIZED POWER VARIATION OF LINEAR FRACTIONAL STABLE MOTIONS
"... ABSTRACT. In this article we deduce a distributional theorem for the realized power variation of linear fractional stable motions. This theorem is proven by choosing the technique of subordination to reduce the proof to a Gaussian limit theorem based on Malliavincalculus. ..."
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ABSTRACT. In this article we deduce a distributional theorem for the realized power variation of linear fractional stable motions. This theorem is proven by choosing the technique of subordination to reduce the proof to a Gaussian limit theorem based on Malliavincalculus.