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.^[1] The theorem is named after Frank Kelly.^[2][3][4][5]

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 non-negative numbers q'_ij and a positive measure π that satisfy the following conditions:^[1]

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

then q'_ij are the rates for the reversed process and π is proportional to the stationary distribution for both processes.

Proof

Given the assumptions made on the q_ij and π we have

${\displaystyle \sum _{i\in S}\pi _{i}q_{ij}=\sum _{i\in S}\pi _{j}q'_{ji}=\pi _{j}\sum _{i\in S}q'_{ji}=\pi _{j}\sum _{i\in S}q_{ji}=\pi _{j},}$

so the

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.

References