A real valued discrete time Markov process {Xn} is defined to be uniformly monotone in the negative (positive) direction if P(x,y)=Pr{XnÅ+1•y•Xn=x}(nP(x,y)=Pr{XnÅ+1≥y•Xn=x}, respectively) is totally positive of order 2 in ¸-•<x,y<•. The uniform monotonicity is a stronger notion than the ordinary stochastic monotonicity. A monotonicity theorem of the same form as Daley’s is first established. Based on this theorem, uniform monotonicity as well as IFR and DFR properties in the Lindley waiting time processes, Markov jump processes and the associated counting processes is discussed. Uniform comparison of (random) sums of random variables is also made. Finally, some applications are given to demonstrate a potential use of this study.
