On the Decrease Rate of the Non-Gaussianness of the Sum of Independent Random Variables

Several proofs of the monotonicity of the nonGaussianness (divergence with respect to a Gaussian random variable with identical second order statistics) of the sum of n independent and identically distributed (i.i.d.) random variables were published. We give an upper bound on the decrease rate of the non-Gaussianness which is proportional to the inverse of n, for large n. The proof is based on the relationship between nonGaussianness and minimum mean-square error (MMSE) and causal minimum mean-square error (CMMSE) in the timecontinuous Gaussian channel.