Maximum Entropy for Sums of Symmetric and Bounded Random Variables: A Short Derivation

Let X₁,..., X_n be n independent, symmetric, random variables on the interval [-1, 1]. Ordentlich (2006) showed that the differential entropy of S_n= Sigma_i=1 ⁿ X_i is maximized when X_i, i = 1,...,n-1 are symmetric Bernoulli random variables and X_n is uniform (-1, 1). We give a short derivation of this result via an alternative proof of a key lemma of Ordentlich (2006).