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.

For a continuous time Markov chain with state space S and transition rate matrix Q (with elements q_ij) if we can find a set of numbers q'_ij and π_i summing to 1 where

then q'_ij are the rates for the reversed process and π_i are the stationary distribution for both processes.

Given the assumptions made on the q_ij and π_i we can see

so the global balance equations are satisfied and the π_i are a stationary distribution for both processes.

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