Notes on the Speed of Entropic Convergence in the Central Limit Theorem

Abstract

In the context of the study of convergence speeds in the Central Limit Theorem, we investigate some consequences of a general Lipschitz contraction property of probability transition kernels with respect to relative entropy. This Markovian approach will enable us to discuss examples whose behavior is not covered by results recently obtained by several authors. More precisely, let X ₀,…, X _n be IID real random variables, centered and normalized. It is known that if their law admits a positive spectral gap and a finite relative entropy with respect to ν, the standard Gaussian distribution, then the relative entropy of law of $({{X}_{0}} + \cdots + {{X}_{n}})/\sqrt {{n + 1}}$ with respect to ν goes to zero at least as $\mathcal{O}(1/(n + 1))$, for large n ∈ ℕ. The two goals of this paper are: on one hand, for any fixed p ∈ ℕ*, to find conditions insuring an entropic convergence faster than $\mathcal{O}(1/{{(n + 1)}^{{p/2}}})$ and on the other hand to relax the spectral gap assumption, even at the cost of slower convergence bounds.