A mixture representation of \pi with applications in Markov chain Monte   Carlo and perfect sampling
  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 
James P. Hobert and Christian P. Robert