Aperiodicity of a Markov Chain

[.....]

Homework Statement

Transition matrix is

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

"argue that this chain is aperiodic"

Homework Equations

definition of aperiodicity - there must exist a time n such that there is a non-zero probability of going from state i to state j for all i & j

The Attempt at a Solution

This definition 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

 

[.....]

anyone?

 

[Mechmathian]

The chain is aperiodic 1->3->2->3->1
You can get from any position to any other (it doesn't have to be in one step..)

 

[.....]

Yeah, I can see it's not periodic and hence must be apeiodic, but what's going on with that definition? My understanding of it is that there has to be a special (fixed) value of n where you can go from anyone state to all the others, including back to that state... but that doesn't seem to hold here... thanks for replying