Entropy and the fourth moment phenomenon

Ivan Nourdin (1) , Giovanni Peccati (2) , Yvik Swan (2)

1 IECL - Institut Élie Cartan de Lorraine
2 FSTC - Faculté des Sciences, de la Technologie et de la Communication

We develop a new method for bounding the relative entropy of a random vector in terms of its Stein factors. Our approach is based on a novel representation for the score function of smoothly perturbed random variables, as well as on the de Bruijn's identity of information theory. When applied to sequences of functionals of a general Gaussian field, our results can be combined with the Carbery-Wright inequality in order to yield multidimensional entropic rates of convergence that coincide, up to a logarithmic factor, with those achievable in smooth distances (such as the 1-Wasserstein distance). In particular, our findings settle the open problem of proving a quantitative version of the multidimensional fourth moment theorem for random vectors having chaotic components, with explicit rates of convergence in total variation that are independent of the order of the associated Wiener chaoses. The results proved in the present paper are outside the scope of other existing techniques, such as for instance the multidimensional Stein's method for normal approximations.

Mots clés: Fourth Moment Theorem; Fisher Information; De Bruijn's Formula; Central Limit Theorem; Carbery-Wright Inequality; Stein factors; Gaussian Fields; Relative Entropy; Stein factors.

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

Ivan Nourdin, Giovanni Peccati, Yvik Swan. Entropy and the fourth moment phenomenon. Journal of Functional Analysis, 2014, 266 (5), pp.3170-3207. ⟨10.1016/j.jfa.2013.09.017⟩. ⟨hal-00807589v2⟩