Annales de l'Institut Henri Poincaré, Probabilités et Statistiques

Fisher information and the fourth moment theorem

Ivan Nourdin and David Nualart

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Using a representation of the score function by means of the divergence operator, we exhibit a sufficient condition, in terms of the negative moments of the norm of the Malliavin derivative under which, convergence in Fisher information to the standard Gaussian of sequences belonging to a given Wiener chaos is actually equivalent to convergence of only the fourth moment. Thus, our result may be considered as a further building block associated to the recent but already rich literature dedicated to the Fourth Moment Theorem of Nualart and Peccati (Ann. Probab. 33 (2005) 177–193). To illustrate the power of our approach, we prove a local limit theorem together with some rates of convergence for the normal convergence of a standardized version of the quadratic variation of the fractional Brownian motion.


À l’aide d’une représentation de la fonction score au moyen de l’opérateur de divergence, nous mettons en évidence une condition suffisante, exprimée en terme de moments négatifs de la norme de la dérivée de Malliavin, sous laquelle la convergence au sens de l’information de Fisher vers la loi gaussienne d’une suite d’éléments appartenant à un chaos de Wiener fixé se trouve être équivalente à la simple convergence du moment quatrième. Nos résultats peuvent être vus comme une nouvelle pierre apportée à l’édification de la récente mais déjà riche littérature dédiée au théorème du moment quatrième de Nualart and Peccati (Ann. Probab. 33 (2005) 177–193). Pour illustrer notre approche, nous prouvons un théorème de la limite locale, avec calcul de la vitesse de convergence associée, pour la convergence normale d’une version renormalisée de la variation quadratique du mouvement brownien fractionnaire.

Article information

Ann. Inst. H. Poincaré Probab. Statist., Volume 52, Number 2 (2016), 849-867.

Received: 20 December 2013
Revised: 9 October 2014
Accepted: 11 October 2014
First available in Project Euclid: 4 May 2016

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60H07: Stochastic calculus of variations and the Malliavin calculus 94A17: Measures of information, entropy 60G22: Fractional processes, including fractional Brownian motion

Fisher information Total variation distance Relative entropy Fourth moment theorem Fractional Brownian motion Malliavin calculus


Nourdin, Ivan; Nualart, David. Fisher information and the fourth moment theorem. Ann. Inst. H. Poincaré Probab. Statist. 52 (2016), no. 2, 849--867. doi:10.1214/14-AIHP656.

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