Stochastic majorization of random variables by proportional equilibrium rates

The equilibrium rate r Y of a random variable Y with support on non-negative integers is defined by r Y (0) = 0 and r Y ( n ) = P [ Y = n – 1] /P [ Y – n ], Let ( j = 1, …, m ; i = 1,2) be 2 m 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