  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.
