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Suppose I have a family of countable state-space, discrete-time Markov chains, indexed by a parameter $r \in \mathbb{R}$. The state space is the same for all values of $r$; the transition probabilities depend on $r$ continuously. Is there some result like:

If the chain obtained for $r=r_0$ is positive recurrent, then the same is true for all values of $r$ in some open neighborhood of $r_0$; and as $r \to r_0$, the associated invariant distributions converge. (I assume that if the first statement is true, then so is the second one.)

In terms of simple linear algebra: I have a countable set of linear equations (defining the invariant distribution in terms of the transition probabilities of the Markov chain), which depend continuously on $r$. At some value $r=r_0$ the system has a unique positive solution. Is there a neighborhood of $r_0$ for which this will still be true? (What conditions do I need to impose on my set of equations to get this?)

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Did you look at Doeblin condition? Don't have a proof, but I think it may help you. –  LazyCat Jan 30 '12 at 23:45

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up vote 4 down vote accepted

No it's not true. Consider Markov chains on $\mathbb N$ where the only transitions are from $n$ to $n\pm 1$.

For $n>1$, set $P_{n,n+1}=r+\frac12(1-\frac1n)$ and $P_{n,n-1}=-r+\frac12(1+\frac1n)$.

For $r=0$, solving the detailed balance equation, we get $\pi_{n+1}n(n+2)=\pi_n(n-1)(n+1)$ which has solutions $\pi_n=C/[(n-1)(n+1)]$. Hence this is positive recurrent.

For $r>0$, there is a rightwards drift and hence it is transient.

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Very nice! But is there a version of the statement that is true? (For example, in my MC all transition rates belong to a finite set, which means that you can't reproduce this construction in it.) –  Elena Yudovina Jan 29 '12 at 8:41
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I doubt that anything much could be said at this kind of level of generality. Even if the transition rates belong to a finite set, I would expect that you could cook up a Markov chain on the integers, where each state could go to 3 other states with probability $p_1(r)$, $p_2(r)$ and $p_3(r)$. The things you would have to play with in an example like this would be $f_1(k)$, $f_2(k)$ and $f_3(k)$ the places you go from state $k$. My guess would be by a clever choice of these guys you could make it barely positive recurrent for $r=0$, but transient for $r>0$. –  Anthony Quas Jan 29 '12 at 8:50

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