Kelly's lemma


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.

Statement

For a continuous time Markov chain with state space S and transition rate matrix Q if we can find a set of numbers qij and πi summing to 1 where
then qij 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.