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 q_ij) if we can find a set of numbers q'_ij and π_i summing to 1 where

${\begin{aligned}\sum _{j\neq i}\pi _{i}q'_{ij}&=\sum _{j\neq i}q_{ij}\quad \forall i\in S\\\pi _{i}q_{ij}&=\pi _{j}q_{ji}'\quad \forall i,j\in S\end{aligned}}$

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

Proof

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

$\sum _{i\neq j}\pi _{i}q_{ij}=\sum _{i\neq j}\pi _{j}q'_{ji}=\pi _{j}\sum _{i\neq j}q_{ji}=-\pi _{j}q_{jj}$

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