The authors consider a discrete-time Markov chain X on the state space 
with stationary one-step transition probabilities such that X is irreducible, transient, aperiodic and skip-free to the left. With 
denoting the modified Markov chain in which the states 
are aggregated into a single absorbing state, they study the conditional state probabilities of 
at time n, given that state 0 will be reached some time after time n. A sufficient condition for the convergence, as 
, of these conditional probabilities to a proper distribution is determined, as well as a condition under which the limiting conditional distribution of X is the limit, as 
, of the limiting conditional distribution of 
For skip-free Markov chains the authors derive a necessary and sufficient condition for the existence of the limiting conditional distribution. As an example of a phenomenon which may be modelled by a limiting conditional distribution, they consider the backlog of a slotted ALOHA protocol.
