On geometric and algebraic transience for discrete-time Markov chains

General characterizations of ergodic Markov chains have been developed in considerable detail. In this paper, we study the transience for discrete-time Markov chains on general state spaces, including the geometric transience and algebraic transience. Criteria are presented through bounding the modified moment of the first return time and establishing the appropriate drift condition. Moreover, we apply the criteria to the random walk on the half line and the skip-free chain on nonnegative integers.