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Suppose $\mu$ is an invariant measure for a Markov chain with state space $S$ with $\mu(i)p_{ij}=\mu(j)p_{ji}$ $\forall i,j \in S$. Describe a Markov chain with this property. Also, show that $\mu$ is an invariant measure.

In constructing the Markov chain, I know I want it to have transition matrix $P$ that satisfies $\mu'=\mu'P$ (prime denoting transpose). I can't think of a named markov chain off the top of my head with this property. Perhaps Gambler's Ruin or deterministically monotone?

To prove it's invariant measure, I need $\sum\limits_{i \in S} \mu(i) \neq 1$. Don't I need the chain before I can do this?

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up vote 1 down vote accepted
  • an example is the transition matrix$\bigl(\begin{smallmatrix} 0&1\\ 1&0 \end{smallmatrix} \bigr)$ with $\mu_1 = \mu_2 = \frac 12$.
  • If $\mu_i p(i,j) = \mu_j p(j,i) $ then summing over $i$ puts $$ \mu_i= \sum_j \mu_i p(i,j) = \sum_j \mu_j p(j,i) = (\mu p)_i $$that is: $\mu$ is invariant.
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How do I know that $(\mu p)_i \neq 1$? – user113525 Mar 13 '14 at 2:01
why do you need this condition? – mookid Mar 13 '14 at 2:02
The definition for invariant measure I was given in lecture said that this is what made it so. – user113525 Mar 13 '14 at 2:04
I have never seen such a condition... for me all there is is: $\mu_i\ge 0$ and $\mu P = \mu$ – mookid Mar 13 '14 at 2:06
I am reading in my textbook now that my lecturer was wrong: $\mu_j=E_i \sum\limits_{0\leq n \leq \tau_i(1)-1} 1_{[X_n=j]}$ qualifies as invariant. What you are saying is stationary distribution (I think). – user113525 Mar 13 '14 at 2:07

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