Cauchy process

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

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

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

$\sum _{i\ne j}{\pi }_i{q}_{ij}=\sum _{i\ne j}{\pi }_{j}q{\text{'}}_{ji}={\pi }_j\sum _{i\ne j}{q}_{ji}=-{\pi }_j{q}_{jj}$

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

References

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