Conditional Fisher-Information Central Limit Theorems under Log-Concavity with Information-Theoretic Consequences

Abstract

We establish conditional central limit theorems in Fisher information under log-concavity in every fixed dimension. For conditionally centered normalized sums, after whitening by the averaged conditional covariance, the averaged conditional Fisher information converges to the dimension if and only if it is finite at one convolution level. The scalar criterion follows as the one-dimensional case; we also provide an independent scalar proof based on a second-order continuity theorem for Fisher production on Gaussian-smoothed, tail-controlled classes. For the original sums, the averaged Fisher information matrix converges in operator norm to the inverse averaged conditional covariance. Consequently, the conditional relative Fisher information with respect to the limiting Gaussian law vanishes, and the Gaussian logarithmic Sobolev inequality yields convergence in conditional relative entropy and conditional entropy. We give two operational consequences. For any fixed finite-constellation low-power input, the first-order conditional mutual-information slope converges to the Gaussian-noise benchmark. For Gaussian signaling at any fixed signal covariance, the mutual-information gap from that benchmark is bounded by the conditional relative Fisher deficit and hence vanishes asymptotically.