## 1. Definition

Let be a Markov chain with transition probabilities .

E.g. A sample realization is 1, 2, 3, 4, 5

Let be the same sequence in reverse, i.e.

Let be the transition probabilities of the reversed process. That is

=

=

=

## 2. Conclusion

In the steady state, assuming the limiting probabilities exist, we have

or

The above equation is saying that the rate of transmissions from i to j in the reversed chain is equal to the rate of transitions from j to i in the forward chain.

A DTMC is time reversible if