  Let X={X_n:n=0,1,2,...} be an irreducible, positive recurrent Markov chain
with invariant probability measure \pi. We show that if X satisfies a one-step
minorization condition, then \pi can be represented as an infinite mixture. The
distributions in the mixture are associated with the hitting times on an
accessible atom introduced via the splitting construction of Athreya and Ney
[Trans. Amer. Math. Soc. 245 (1978) 493-501] and Nummelin [Z. Wahrsch. Verw.
Gebiete 43 (1978) 309-318]. When the small set in the minorization condition is
the entire state space, our mixture representation of \pi reduces to a simple
formula, first derived by Breyer and Roberts [Methodol. Comput. Appl. Probab. 3
(2001) 161-177] from which samples can be easily drawn. Despite the fact that
the derivation of this formula involves no coupling or backward simulation
arguments, the formula can be used to reconstruct perfect sampling algorithms
based on coupling from the past
