Entropic conditional central limit theorem and Hadamard compression

The Hadamard compression has been proved to achieve the compression limit for mixed-distributed input signals, by making the uncertainty of the discrete part vanish. The limit of the continuous part is still unknown. We find that the Hadamard transform is closely related to the conditional central limit theorem since they both satisfy a core entropic property. We first establish an entropic conditional central limit theorem (CCLT), which is stronger than the classical CCLT. Second, through building a suitable probability space and extending the entropic CCLT, we show that for the continuous input under the iterated Hadamard transform, almost every distribution of the output conditional on the values of the previous signals will tend to be Gaussian, and the conditional distribution is in fact insensitive to the condition. The results enable us to make a theoretical study concerning the Hadamard compression, which provides a solid theoretical analysis supporting the simulation results in a previous work.