On first returning time and last exit time of a class of Markov chain

Let {X n } be a Markov chain with transition probability pij=aj.(i.1)+,.i,j.0, where a j = 0 provided j < 0, a 0 > 0, a 0 + a 1 < 1 and . .n=0 a n = 1. Let µ = . .n=1 na n . It is known that {X n } is positive recurrent when µ < 1; is null recurrent when µ = 1; and is transient when µ > 1. In this paper, the integrability of the first returning time and the last exit time are discussed.