  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.
