Monotonicity of Entropy and Fisher Information: A Quick Proof via   Maximal Correlation
  A simple proof is given for the monotonicity of entropy and Fisher information associated to sums of i.i.d. random variables. The proof relies on a characterization of maximal correlation for partial sums due to Dembo, Kagan and Shepp. 
Thomas A. Courtade
cs.IT math.IT math.PR