Aperiodicity of a Markov Chain

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Homework Help Overview

The discussion revolves around the aperiodicity of a Markov chain characterized by a given transition matrix. Participants are examining the definition of aperiodicity and its implications for the states within the chain.

Discussion Character

  • Conceptual clarification, Assumption checking

Approaches and Questions Raised

  • The original poster attempts to understand the definition of aperiodicity and questions whether their interpretation aligns with the behavior of the transition matrix. Other participants explore the chain's transitions and suggest that movement between states can occur over multiple steps.

Discussion Status

Participants are actively engaging with the definition of aperiodicity and its application to the specific Markov chain. Some have provided insights into the transitions, while others are seeking clarification on the definition itself. There is no explicit consensus, but the discussion is progressing with various interpretations being explored.

Contextual Notes

There appears to be confusion regarding the definition of aperiodicity, particularly in relation to the requirement of a fixed value of n for transitions between states. This uncertainty is central to the discussion.

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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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anyone?
 
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
 

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