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I got a quick question I got a markov chain with this trans matrix $$\begin{pmatrix}1&0\\1/2&1/2\end{pmatrix}$$

And I got 2 states right [0,1] right. So I know state 0 has a period 1 and is positive recurrent cause it keep returning to itself but has period 1.

But not sure about state 1. I know it has period 1 because it keep returning to itself. But I think it also transient because once it leave for 0 it cannot come back. So would it be period 0 or period 1. Cause I guess you could make a choice to never leave state 1.

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State 1 is indeed transient: when starting from state 1 there is a positive probability that you never return to state 1 (in this case, because you are stuck in state 0).

The period of a state $i$ is defined as $\gcd\{n \geq 1 : p_{i,i}(n) > 0\}$, where $\gcd$ denotes the greatest common divisor and $p_{i,i}(n)$ denotes the probability of being in state $i$ after $n$ steps. Thus, in this example, both states have period 1. A state with period 1 is also called aperiodic.

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