We make use of an entropic property to establish a convergence theorem (Main Theorem), which reveals that the conditional entropy measures the asymptotic Gaussianity. As an application, we establish the {\it entropic conditional central limit theorem} (CCLT), which is stronger than the classical CCLT. As another application, we show that continuous input under iterated Hadamard transform, almost every distribution of the output conditional on the values of the previous signals will tend to Gaussian, and the conditional distribution is in fact insensitive to the condition. The results enable us to make a theoretic study concerning Hadamard compression, which provides a solid theoretical analysis supporting the simulation results in previous literature. We show also that the conditional Fisher information can be used to measure the asymptotic Gaussianity.

翻译：我们利用一个熵性质建立了收敛定理（主定理），该定理揭示了条件熵可度量渐近高斯性。作为一个应用，我们建立了《熵条件中心极限定理》（CCLT），该定理比经典CCLT更强。作为另一个应用，我们证明在迭代哈达玛变换下，连续输入信号的条件输出分布（基于先前信号值）几乎必然趋于高斯分布，且该条件分布实际上对条件不敏感。这些结果使我们能够对哈达玛压缩进行理论研究，为先前文献中的仿真结果提供坚实的理论分析支撑。我们还证明条件Fisher信息可用于度量渐近高斯性。