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If it's irreudible, then you can "get to anywhere from anywhere." a transient state is one that eventually leaves and never returns. but doesn't this contradict the irreducibility premise that you can get to anywhere from anywhere?

beginner
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    There's no contradiction. For example, a non-symmetric random walk is both irreducible and transient. – zhoraster Mar 14 '22 at 20:06

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There's a difference between "can go" and "will go".

For a Markov chain $(X_n)$ write $T_y:=\inf\{n\ge 1: X_n=y\}$ and $P^x$ for the law of $(X_n)$ started at $X_0=x$.

Irreducible: $P^x[T_y<\infty]>0$ for all $x,y$.

Suppose $(X_n)$ is irreducible. Then $(X_n)$ is

Recurrent if $P^x[T_x<\infty]=1$ for all $x$;

Transient if $P^x[T_x<\infty]<1$ for all $x$.

John Dawkins
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