In probability theory, Kelly's lemma states that for a stationary continuous time Markov chain, a process defined as the time-reversed process has the same stationary distribution as the forward-time process. The theorem is named after Frank Kelly.
Contents
Statement
For a continuous time Markov chain with state space S and transition rate matrix Q (with elements qij) if we can find a set of numbers q'ij and πi summing to 1 where
then q'ij are the rates for the reversed process and πi are the stationary distribution for both processes.
Proof
Given the assumptions made on the qij and πi we can see
so the global balance equations are satisfied and the πi are a stationary distribution for both processes.
References
Kelly's lemma Wikipedia(Text) CC BY-SA