Stochastic majorization of random variables by proportional equilibrium rates

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], (n >= 1).Let Yj(i), (j = 1, ., m; i = 1,2) be 2m independent random variables that have proportional equilibrium rates with (1/rho j(i)), (j = 1, ., m; i = 1, 2) as the constant of proportionality. When the equilibrium rate is increasing and concave [convex] it is shown that rho(1)=rho1(1), ., rhom(1)) majorizes rho(2)=rho1(2), ..., rhom(2) implies Eg(Y1(1), ., Ym(1)>= [<=] Eg(Y1(2), ..., Ym(2)) for all increasing Schur-convex [concave] functions g: Rm --> R, whenever the expectations exist. In addition if rho1(i)>=rho2(i)>=...>=rhom(i), (i = 1, 2), then https://static.cambridge.org/content/id/urn%3Acambridge.org%3Aid%3Aarticle%3AS0001867800017468/resource/name/S0001867800017468_eq1.gif?pub-status=live