Usual and stochastic tail orders between hitting times for two Markov chains

We consider time-homogeneous Markov chains with state space Ek≡{0,1,…,k} and initial distribution concentrated on the state 0. For pairs of such Markov chains, we study the Stochastic Tail Order and the stochastic order in the usual sense between the respective first passage times in the state k. On this purpose, we will develop a method based on a specific relation between two stochastic matrices on the state space Ek. Our method provides comparisons that are simpler and more refined than those obtained by the analysis based on the spectral gaps.