Period of a Markov Chain: Why is this one aperiodic?

Question

Here is the problem from a stochastic processes book:

Consider a Markov Chain on {0,1,2} having transition matrix

   0  1  2
0| 0  0  1|
1| 1  0  0|
2|.5 .5  0|

The answer in the book states that the period is 1, and therefore the chain is aperiodic. But I do not understand why the period is not 2.

The quickest way to return to any state is in 2 steps,

0 --> 2 --> 0

So shouldn't the chain be periodic with a period of 2?

Any explanation would be appreciated.

Answer 1

Double check the definition of the period. First off, a period always refers to a particular state $i$. Specifically it's the GCD of all times $k$ for which a return is possible. So figuring out the "shortest time of return" is not sufficient.

Next, a state $i$ is aperiodic if its period is 1. A Markov Chain is aperiodic if all states have period 1.

In your example, it's possible to start at 0 and return to 0 in 2 or 3 steps, therefore 0 has period 1. Similarly, 1 and 2 also have period 1. So the Markov chain is aperiodic.