Aperiodicity of a Markov Chain

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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
 
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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
 
There are two things I don't understand about this problem. First, when finding the nth root of a number, there should in theory be n solutions. However, the formula produces n+1 roots. Here is how. The first root is simply ##\left(r\right)^{\left(\frac{1}{n}\right)}##. Then you multiply this first root by n additional expressions given by the formula, as you go through k=0,1,...n-1. So you end up with n+1 roots, which cannot be correct. Let me illustrate what I mean. For this...
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