Aperiodicity of a Markov Chain

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The discussion revolves around determining the aperiodicity of a given Markov chain represented by a specific transition matrix. Aperiodicity is defined as the existence of a time n where there is a non-zero probability of transitioning between any two states for all i and j. The original poster expresses confusion, noting that transitions from state 1 to state 2 require odd n, while transitions to states 1 and 3 require even n. However, responses clarify that aperiodicity does not require a fixed n for all transitions, as long as it is possible to reach any state from any other state over time. Ultimately, the chain is deemed aperiodic due to the ability to transition between all states, albeit not necessarily in a fixed number of steps.
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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
 

Thread 'Does this series converge uniformly?'

First, I tried to show that ##f_n## converges uniformly on ##[0,2\pi]##, which is true since ##f_n \rightarrow 0## for ##n \rightarrow \infty## and ##\sigma_n=\mathrm{sup}\left| \frac{\sin\left(\frac{n^2}{n+\frac 15}x\right)}{n^{x^2-3x+3}} \right| \leq \frac{1}{|n^{x^2-3x+3}|} \leq \frac{1}{n^{\frac 34}}\rightarrow 0##. I can't use neither Leibnitz's test nor Abel's test. For Dirichlet's test I would need to show, that ##\sin\left(\frac{n^2}{n+\frac 15}x \right)## has partialy bounded sums...
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