Is the Given Markov Chain Aperiodic?

[.....]

my transition matrix is

0 0 1
0 0 1
(1/3) (2/3) 0

I'm supposed to argue that this chain is aperiodic,

A markov chain is aperiodic iff there exists a time n such that there is a positive probability of going from state i to state j for all i and j

This doesn't seem to hold for my chain ... for example, to go from state 1 to state 2 n has to be odd.. but to go from state 1 to state 1 or 3 n has to be even..

Am I just getting this definition muddled up? Could someone elaborate on it for me? Thanks

 

[Valkarie]

in advance!Yes, you are getting this definition muddled up. The definition of aperiodicity is that for all states i and j, there exists an n such that the probability of transitioning from i to j is positive. This means that it doesn't matter whether n is odd or even, there must exist some n such that the probability is positive. In your transition matrix, we can see that all states have a positive probability of transitioning to each other state. For example, the probability of transitioning from state 1 to state 2 is 0, but the probability of transitioning from state 1 to state 3 is 1/3 (which is positive). Similarly, the probability of transitioning from state 2 to state 3 is 2/3 (which is positive). Therefore, since all states have a positive probability of transitioning to each other, this chain is aperiodic.