Kelly's lemma

Template:Short description 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 an at most countable state space $S$ and transition-rate matrix $Q$ (with elements $q_{ij}$), if we can find a set of non-negative numbers $q_{i^{\prime }j}$ and a positive measure $\pi$ that satisfy the following conditions:

$\begin{array}{cc}\sum _{j\in S}{q}_{ij}& =\sum _{j\in S}q{\text{'}}_{ij}\mathrm{\forall }i\in S\\ {\pi }_i{q}_{ij}& ={\pi }_j{q}_{j^{\prime }i}\mathrm{\forall }i,j\in S,\end{array}$

then $q_{i^{\prime }j}$ are the rates for the reversed process and $\pi$ is proportional to the stationary distribution for both processes.

Proof

Given the assumptions made on the $q_{ij}$ and $\pi$ we have

$\sum _{i\in S}{\pi }_i{q}_{ij}=\sum _{i\in S}{\pi }_{j}q{\text{'}}_{ji}={\pi }_j\sum _{i\in S}q{\text{'}}_{ji}={\pi }_j\sum _{i\in S}{q}_{ji}={\pi }_j,$

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