Concavity of entropy under thinning

Building on the recent work of Johnson (2007) and Yu (2008), we prove that entropy is a concave function with respect to the thinning operation T_alpha. That is, if X and Y are independent random variables on Z₊ with ultra-log-concave probability mass functions, then H(T_alphaX + T_1-alphaY) ges alphaH(X) + (1 - alpha)H(Y), 0 les alpha les 1, where H denotes the discrete entropy. This is a discrete analogue of the inequality (h denotes the differential entropy) h(radicalphaX + radic1 - alphaY ) ges alphah(X) + (1 - alpha)h(Y), 0 les alpha les 1, which holds for continuous X and Y with finite variances and is equivalent to Shannon´s entropy power inequality. As a consequence we establish a special case of a conjecture of Shepp and Olkin (1981). Possible extensions are also discussed.