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Let $X$ be a discrete-time Markov process in $S$ with invariant distribution $\nu$. Show that for any measurable set $B\subset S$ such that $$P_{\nu}\{X_n \in B\, \textrm{i.o.} \}\geq \nu B.$$

I'm honestly also unsure what $P_{\nu}$ means here. Is that supposed to be the initial distribution (e.g. the distribution of $X_0$)? If that's the case, I guess I can prove this using a series argument for the LHS.

Kashif
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1 Answers1

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The measure $P_\nu$ specifies that the initial distribution is $\nu$. Therefore $$ P_\nu [A] = \sum_{i \in S} \nu(i) P_i[A] = \sum_{i \in S} \nu(i) P[A \mid X_0 = i] $$ for any measurable set $A$.

Notice that if $\nu$ is the invariant distribution for the chain then $$ P_\nu[ X_n \in B \text{ i.o.}] = P_\nu[ \limsup \{X_n \in B \}] \geq \limsup P_\nu[X_n \in B] = \sum_{i \in S} \nu(i) \sum_{j \in B} K_{ij}^n $$ where $K$ is the kernel for the chain. Since $\nu$ is a left evector for $K$ with evalue $1$ (this is what it means to be invariant) the last term above is $\nu(B)$.

user139388
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  • How would this imply recurrence for a state $z$ if $\nu{z} > 0$? – Kashif Apr 09 '14 at 12:45
  • I'm trying to find a way to do this using Fatou's lemma but I don't see why that would be needed because that deals with lim inf whereas here, we have an expression involving limsup (X_n i.o.). – Kashif Apr 09 '14 at 14:05
  • Hey! Your question didn't say anything about recurrence, but to show recurrence of $i$ you need $\sum_{n = 0}^\infty P_i[X_n = i] = \infty$. If your chain is ergodic then the terms of this series are converging to $\nu(i)$, a non-zero constant, and therefore the series diverges.

    Why do you want to use Fatou's lemma? Also, were you okay with the argument I gave? Thanks!

     – user139388 Apr 09 '14 at 20:24
  • Yeah thanks! This was perfect. – Kashif Apr 10 '14 at 22:09