We consider a discrete-time Markov chain X on the state space ℕ ≡ {0, 1, ...} with stationary one-step transition probabilities such that X is irreducible, transient, aperiodic and skip-free to the left. With X(m) denoting the modified Markov chain in which the states {m, m + 1, ...} are aggregated into a single absorbing state, we study the conditional state probabilities of X (resp. X(m)) at time n, given that state 0 will be reached some time after time n. A sufficient condition for the convergence, as n → ∞, 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 m → ∞, of the limiting conditional distribution of X(m). For skip-free Markov chains we 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, we consider the backlog of a slotted ALOHA protocol.
