AI 中文总结
本文证明对数凹性下条件Fisher信息中心极限定理，并由此推出条件熵收敛及互信息斜率趋于高斯基准等推论。
AI 中文摘要
我们在任意固定维度下，建立了对数凹性条件下关于Fisher信息的条件中心极限定理。对于条件中心化后的归一化和，在通过平均条件协方差进行白化后，平均条件Fisher信息收敛到维度当且仅当它在一个卷积层上是有限的。标量准则作为一维情形随之得出；我们还基于高斯平滑、尾部受控类上Fisher产生的二阶连续性定理，给出了一个独立的标量证明。对于原始和，平均Fisher信息矩阵在算子范数下收敛到平均条件协方差的逆。因此，相对于极限高斯分布的条件相对Fisher信息消失，并且高斯对数Sobolev不等式导出条件相对熵和条件熵的收敛性。我们给出两个操作性的推论。对于任何固定的有限星座低功率输入，一阶条件互信息斜率收敛到高斯噪声基准。对于任何固定信号协方差的高斯信号，与基准的互信息差距受条件相对Fisher亏缺限制，因此渐近消失。
英文摘要
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.
Comments47 pages
