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Suppose I have an infinite and reversible Markov chain $X_n$ with transition kernel $P_1(x,y)$ and stationary measure $\pi(x)$.

For concreteness, suppose my state space is on a graph and my edges have weights $a_{xy}$, so that $$P_1(x,y)=\frac {a_{xy}}{\sum_{z\sim x}a_{xz}}$$ I'm interested in references which deal with the following:

Suppose $a_{x^*y^*}$ (the weight between vertices $x^*$ and $y^*$) is changed to $b_{x^*y^*}$. In particular, by changing a single weight, I get a new transition kernel $P_2(x,y)$.

What can be said about the behavior of $P^n_2(x,x)$ compared to $P^n_1(x,x)$ as $n$ gets large?

If I tell you that every state in this Markov chain is transient, can something be said about the Radon Nikodym derivative $dP_1/dP_2$, in the sense that $\displaystyle P_1^n(x,x)=E_x^{(2)}\left(\frac{dP_1}{dP_2} 1_{X_n=x}\right)$.

I would like to say that $P_1^n(x,x)$ and $P_2^n(x,x)$ become commensurate but, this likely needs strong conditions on what $b_{x^*y^*}$ can be.

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I fixed the tilde, but I don't know what you mean by the subscripted $1$. Is that supposed to be a vertical bar? What do you mean by $X_n=0$? –  joriki May 28 '11 at 4:35
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1 Answer 1

Even after some rewriting (whose conformity with the OP's intent needs validation) by joriki and myself, the question is not absolutely clear. Below is one interpretation, in which the reversibility and transience of the Markov chain are simply not relevant.

To wit, it seems one is left with the hypothesis that $P_1$ and $P_2$ are two Markov transition kernels such that there exists a finite and positive constant $c$ with, for every states $x$ and $y$, $$c^{-1}P_1(x,y)\le P_2(x,y)\le cP_1(x,y).$$ Then it is a general (and easy) fact that, for every $n\ge1$ and state $x$, $$c^{-n}P_1^n(x,x)\le P_2^n(x,x)\le c^nP_1^n(x,x).$$ If $n$ and $P_1$ are fixed and one chooses different kernels $P_2$ more and more like $P_1$ in the sense that $c\to1$, then obviously $P_2^n(x,x)$ is more and more like $P_1^n(x,x)$.

In the reversible situation described by the OP where one changes the weight $a_{x^*y^*}$ of a single edge $x^*y^*$, $P_1(x,y)$ is modified only when $x$ or $y$ is $x^*$ or $y^*$ but it is not clear how this helps. Of course, if $b_{x^*y^*}\to a_{x^*y^*}$, then $c\to1$. Quantitative estimates on $c$ could be derived from a more precise description of the deformation of $a$ into $b$.

Finally, note that the Radon-Nykodym derivative $dP_1/dP_2$ invoked by the OP does not exist since $P_1$ and $P_2$ are singular to each other. But, for every fixed time $n$, there exists a Radon-Nykodym derivative of the restrictions of $P_1$ and $P_2$ to the sigma-algebra $F_n$ of the past of the Markov chain up to time $n$.

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