Update: Oops! This is a stupid question and should be closed. The definition of the probability space that contains events $A_i$ requires using a single random stream.

I have difficulties understanding the proof of the Propp-Wilson algorithm in the book "Finite Markov Chains and Algorithmic Applications" 1 by Olle Häggström. I'm not sure how the "reusing old randomness" is used in the proof given in the book.

The proof is on page 79, below the theorem 10.1. It goes as follows:

My question is: Where is the use of "reusing old randomness" in the proof? I intuitively understand that if one generates completely new random numbers in each round, the algorithm will be more likely to over-sample those paths that couple together quickly. Hence the sampling is biased. However, I can't see how this is used in the proof.

1. http://cms.dm.uba.ar/academico/materias/verano2016/probabilidades_y_estadistica_C/Haggstrom-Finite%20Markov%20chains%20and%20algorithm%20applications.pdf