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So i had an exam returned today, and im not convinced that my answer was wrong so it went like this.

Markovian Chain Picture

The question was if the the markov chain had a period (The correct answer is it has a period). And me looking at it, i was sure there was no period, but my classmates said that state E had a period of 0, Im not convinced with their answer, so I searched over the internet and i cant find samples of markov chains with period 0 because first of all it doesnt make sense, having a period would mean that the markov chain would return to the same state over some time, but if E had a period of 0 that would mean no transition is required for the state to return to E, but that is clearly not the case. Also the markov chain cannot have a period anywhere else, since there are many recurrent classes and even absorbing classes (period = 1 doesnt count because by definition period = 1 is aperiodic).

so i was thinking that maybe I overlooked something else, Im asking you guys which state here has a period

the teacher left early so i wasnt able to ask him about this so im asking you guys instead.

Wuh
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  • @JohnD It's not clear to me what your question is. Do you want to know why the Markov chain in this question has a period, or do you actually want to know the question in the title (no, "period zero" makes no sense)? – Misha Lavrov May 08 '22 at 16:18
  • "Can a markov chain have period of zero?" is my question, I'm curious if there anything to that end. Probably not since period only exists for irreducible classes – John D May 08 '22 at 21:26

2 Answers2

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Any Markov chain with a state that has a self-loop is aperiodic.

user365239
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The period of a markov chain is d = $hcf\{n~|~f_n >0\}$, where $f_n = p(T_i =n)$ is the probability of returning to a particular state in $n$ steps. Hence, it always has a period, it's just that if $d = 1$, we refer to it as aperiodic. But it still has a period, by definition.

Cai
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