Let be a probability distribution on the nonnegative integers such that $\pi_i > 0$ for all i. Write down the transition matrix of an irreducible, aperiodic, recurrent Markov chain on the nonnegative integers that has as its stationary probability distribution.
File can be found here: https://tbp.berkeley.edu/examfiles/stats/stats150-sp07-final-Evans-exam.pdf (#1 on this site)
What I notice is that I am trying to reduce it to a finite case, then go from specific to general. Maybe I can try the geometric distribution but I not really seeing how using the detailed balance equations here. Many Thanks