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This question is related to a comment in Rick Durret's book, Probability: Theory and Examples (version 5). In the proof for Theorem 5.5.9, which is about the uniqueness of stationary measure for irreducible and recurrent Markov chains with a countable state space $S$. It's stated that for $a \in S$, $P_\nu(X_j \neq a, 0 \le j <n)$ might not go to zero if $\nu$ is an infinite measure. I don't understand why this is true, and how to show it mathematically.

Note that here $\nu$ is a stationary measure, $P_\nu$ is a probability measure on the path space where the initial measure is $\nu$.

statstats
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  • What is meant by "an infinite measure"? (In this context) – Michael Mar 29 '24 at 18:33
  • I think it means $\nu(S)=\infty$. – statstats Mar 30 '24 at 10:08
  • Yes, but in this context, I do not know what $P_v$ means, it seems to be mixing probability measures and general measures. For Markov chains, usually $P_v[X_k=i]$ is defined as $P[X_k=i|X_0=v]$, so I would expect $v$ to be a state in $S$. Now if $v$ is meant to be a distribution then I would expect it to be a probability distribution on the initial state $X_0 \in S$. So $v(S)=1$. I cannot make sense out of $v(S)=\infty$ here: In what sense can a probability measure $P_v$ be parameterized by an infinite measure $v$? What is the assumption on the initial state $X_0$? – Michael Mar 30 '24 at 14:50
  • Yes I also find it very confusing. In this book, the notations are defined as: for $x \in S$, $P_x$ denotes the probability measure on the path space with $X_0=x$, or for a distribution $\mu$, $P_\mu$ denotes the probability measure on the path space with initial distribution $\mu$. So I'm also not sure why here $\nu$ can be an infinite measure ... I guess conceptually, it means the process can start from any state in $S$ with equal probability. – statstats Mar 30 '24 at 23:18
  • I was also wondering whether here maybe the author only wanted to make a conceptual comment without math rigorousness. However, after this comment, the book concludes that because of this possibility of infinite measure $\nu$, we have to use a different way to show $P_\nu(X_j \neq a, 0 \le j <n) \to 0$ as $n \to \infty$. So I felt like this isn't just a casual comment ...therefore trying to figure out how to prove it. – statstats Mar 30 '24 at 23:25

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