A Finite-Entropy Criterion for the Entropic Conditional Central Limit Theorem

Source abstract

We prove a finite-entropy criterion for the entropic conditional central limit theorem. Let $(ξ_i,η_i)_{i\geq 1}$ be independent copies of a pair $(ξ,η)$, and set $W_n=n^{-1/2}\sum_{i=1}^n ξ_i$ and $\boldsymbolη_n=(η_1,\ldots,η_n)$. Under the assumptions that $\mathbb{E}\operatorname{Var}(ξ\midη) -\infty$ for some $n_0$. The main technical ingredient is a continuity theorem for Fisher information under Gaussian smoothing, which allows us to replace the finite expected conditional Fisher-information assumption by a necessary and sufficient finite-entropy condition.