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 (with elements qij) if we can find a set of non-negative numbers q'ij and a positive measure π that satisfy the following conditions:
 * $$\begin{align}

\sum_{j \in S} q_{ij} &= \sum_{j \in S} q'_{ij} \quad \forall i\in S\\ \pi_i q_{ij} &= \pi_jq_{ji}' \quad \forall i,j \in S, \end{align}$$ then q'ij are the rates for the reversed process and π is proportional to the stationary distribution for both processes.

Proof
Given the assumptions made on the qij and π we have
 * $$ \sum_{i \in S} \pi_i q_{ij} = \sum_{i \in S} \pi_j q'_{ji} = \pi_j \sum_{i \in S} q'_{ji} = \pi_j \sum_{i \in S} q_{ji} =\pi_j,$$

so the global balance equations are satisfied and the measure π is proportional to the stationary distribution of the original process. By symmetry, the same argument shows that π is also proportional to the stationary distribution of the reversed process.