Stochastic majorization of random variables by proportional equilibrium rates

Abstract

The equilibrium rate rY of a random variable Y with support on non-negative integers is defined by rY(0) = 0 and rY(n) = P[Y = n – 1]/P[Y – n], Let (j = 1, …, m; i = 1,2) be 2m independent random variables that have proportional equilibrium rates with (j = 1, …, m; i = 1, 2) as the constant of proportionality. When the equilibrium rate is increasing and concave [convex] it is shown that , …, ) majorizes implies , …, for all increasing Schur-convex [concave] functions whenever the expectations exist. In addition if , (i = 1, 2), then