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

Abstract

We prove a finite-entropy criterion for the entropic conditional central limit theorem. Let $({\xi }_i,{\eta }_i{)}_{i\ge 1}$ be independent copies of a pair $(\xi ,\eta )$, and set $W_n=n^{-1/2}{\sum }_{i=1}^n{\xi }_i$ and ${\mathbf{\eta }}_n=({\eta }_1,\dots ,{\eta }_n)$. Under the assumptions that $\mathbb{E}\mathrm{Var}(\xi \mid \eta )<\infty$ and that the conditional law of $\xi$ given $\eta$ is absolutely continuous almost surely, we show that $\mathbb{E}h(W_n\mid {\mathbf{\eta }}_n)$ converges to the Gaussian entropy $\frac{1}{2}\log (2\pi e{\sigma }^2)$, where ${\sigma }^2=\mathbb{E}\mathrm{Var}(\xi \mid \eta )$, if and only if $\mathbb{E}h(W_{n_0}\mid {\mathbf{\eta }}_{n_0})>-\infty$ 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.

12 pages. Submitted to Electronic Communications in Probability