On an Inequality of Karlin and Rinott Concerning Weighted Sums of i.i.d. Random Variables

It is convenient to define H(X) = Hα(X) = −∞ when X is discrete, e.g., degenerate. (Our notation differs from that of Karlin and Rinott 1981 here.) We study the entropy of a weighted sum, S = ∑n i=1 aiXi, of i.i.d. random variables Xi, assuming that the density f of Xi is log-concave, i.e., supp(f) = {x : f(x) > 0} is an interval and log f is a concave function on supp(f). The main result is that H(S) (or Hα(S) with 0 < α < 1) is smaller when the weights a1, . . . , an are more “uniform” in the sense of majorization. A real vector b = (b1, . . . , bn) ⊤ is said to majorize a = (a1, . . . , an) , denoted a ≺ b, if there exists a doubly stochastic matrix T , i.e., an n × n matrix (tij) where tij ≥ 0, ∑