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

TL;DR

This paper establishes an upper bound on how quickly the non-Gaussianness of the sum of i.i.d. random variables decreases as the number of variables increases, linking it to MMSE and CMMSE in Gaussian channels.

Contribution

It provides a novel upper bound on the decrease rate of non-Gaussianness, enhancing understanding of the convergence to Gaussianity for sums of i.i.d. variables.

Findings

Upper bound proportional to 1/n for large n

Connection between non-Gaussianness and MMSE/CMMSE

Improved understanding of Gaussian convergence rate

Abstract

Several proofs of the monotonicity of the non-Gaussianness (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 non-Gaussianness and minimum mean-square error (MMSE) and causal minimum mean-square error (CMMSE) in the time-continuous Gaussian channel.