Fisher information and the Fourth Moment Theorem

Ivan Nourdin (1) , David Nualart (2)

1 IECL - Institut Élie Cartan de Lorraine
2 Department of Mathematics [Lawrence, Kansas]

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. 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.

Mots clés: Fractional Brownian motion; Malliavin calculus; Fourth Moment Theorem; relative en tropy; total variation distance; Fisher information

Domaines: Probabilités [math.PR]; Théorie de l'information et codage [math.IT]; Théorie de l'information [cs.IT]

Ivan Nourdin, David Nualart. Fisher information and the Fourth Moment Theorem. 2013. ⟨hal-00921171⟩