  This article presents a new proof of the rate of convergence to the normal
distribution of sums of independent, identically distributed random variables
in chi-square distance, which was also recently studied in \cite{BobkovRenyi}.
Our method consists of taking advantage of the underlying time non-homogeneous
Markovian structure and studying the spectral properties of the non-reversible
transition operator, which allows to find the optimal rate in the convergence
above under matching moments assumptions. Our main assumption is that the
random variables involved in the sum are independent and have polynomial
density, interestingly, our approach allows to relax the identical distribution
hypothesis.
