The coupling from the past algorithm is a way of perfect sampling from the stationary distribution of irreducible, periodic, and positive recurrent Markov Chain. The algorithm is based on a random graph called the Deoblin Graph. The Doeblin Graph of a countable state space Markov chain describes the joint pathwise evolutions of the Markov dynamics starting from all possible initial conditions, with two paths coalescing when they reach the same point of the state space at the same time. Its Bridge Doeblin subgraph only contains the paths starting from a tagged point of the state space at all possible times. In the irreducible, periodic, and positive recurrent case, the properties of the Bridge Doeblin Graph are known in the literature.
In this talk, the properties of the Bridge Doeblin Graph will be discussed when it is constructed by a null recurrent Markov Chain. As a result, a definition for the perfect sampling of stationary measures of null recurrent Markov Chains will be introduced.