  We introduce the concept of asymptotic period for an irreducible and
aperiodic, discrete-time Markov chain X on a countable state space, and develop
the theory leading to its formal definition. The asymptotic period of X equals
one - its period - if X is recurrent, but may be larger than one if X is
transient; X is asymptotically aperiodic if its asymptotic period equals one.
Some sufficient conditions for asymptotic aperiodicity are presented. The
asymptotic period of a birth-death process on the nonnegative integers is
studied in detail and shown to be equal to 1, 2 or infinity. Criteria for the
occurrence of each value in terms of the 1-step transition probabilities are
established.
