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.

## Contents

## 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

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

so the global balance equations are satisfied and the *π*_{i} are a stationary distribution for both processes.

## References

Kelly's lemma Wikipedia(Text) CC BY-SA