Let   be a Markov chain with state space in R₊ = (0,∞), an initial distribution μ and a transition probability Q. For each x∈R₊ the support of   is [0,x], which implies that  . Set   and put  . We prove that   is a Markov renewal process and that   is a Markov process with a stationary transition probability function. Write   and suppose that  . We give conditions under which   is relatively stable and show that  , where   are stabilizing constants and Z is exponentially distributed. We also show that   is asymptotically stationary and possesses a mixing property