Girish Mahajan (Editor)

Kelly's lemma

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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 qij) if we can find a set of numbers q'ij and πi summing to 1 where

then q'ij are the rates for the reversed process and πi are the stationary distribution for both processes.

Proof

Given the assumptions made on the qij and πi we can see

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

References

Kelly's lemma Wikipedia


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