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Let $P(x,A)$ be a stochastic kernel on a measurable space $(E,\mathcal E)$ and $G = \sum\limits_0^\infty P^n$ be its potential kernel. A $\sigma$-finite measure $\phi$ is called the irreducibility measure if $\phi(A)>0$ implies $PG(x,A)>0$ for all $x\in E$.

In Numellin's book it is written that if $\phi$ is an irreducibility measure then so is $\phi P$. I've tried to prove this fact but didn't manage to do it, so I hope you can help me with an idea.


Just to clarify things: kernel $P$ means that $P(x,\cdot)$ is a probability measure on $(E,\mathcal E)$ for all $x$ and $P(\cdot, A)$ is a measurable function for all $E$. The product of kernels, say $P$ and $G$ is $$ PG(x,A) = \int\limits_E G(y,A)P(x,\mathrm dy) $$ and $\phi P(A) = \int\limits_E P(y,A)\phi(\mathrm dy)$.

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Well, $\phi P(A) > 0$ implies $\exists B\subset E: \phi(B)>0$ and $P(y,A)>0\ \forall y\in B$. This implies by definition of $\phi$ that $PG(x,B) > 0$ for all $x\in E$, whence $P^2G(x,A) = \int P(y,A) PG(x,dy) > 0$ for all $x\in E$. But $PG(x,A) \ge P^2G(x,A)$.

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Thank you very much, Pascal. Correct me if I wrong: we can take $B = \{y:P(y,A) > 0\}$ and so $$ P^2 G(x,A) = \int\limits_E P(y,A)PG(x,dy) = \int\limits_B P(y,A)PG(x,dy) >0 $$ since $PG(x,B)>0$ for all $x\in E$ from the fact that $\phi(B)>0$. If I may ask - how did you get the intuition about this proof? I feel not very confident in dealing with closed sets and irreducibility so far, so in some problems for me it's hard to get the idea. –  Ilya Apr 27 '12 at 12:57
    
@Ilya, you are welcome. Yes, you can define $B$ like this. For the intuition: $\phi$ "gives" the set of all states that can be reached from any point. Now if from these states we take one step further, i.e. if we map $\phi$ to $\phi P$, then a fortiriori, we can reach these states from any point. –  Pascal Maillard Apr 28 '12 at 6:58
    
@Ilya, feel free to accept the answer, if it correctly answered your question. –  Pascal Maillard Apr 28 '12 at 7:09
    
@Pascal: I've just accepted it - just wanted first to verify the answer by myself to be sure that I've understood in completely. I wonder if you can advise any references with exercises on the topic of $\phi$-irreducibility. Or maybe even references on the application of this theory to the analysis of the concrete models. –  Ilya Apr 28 '12 at 18:48
    
@Ilya, thanks for accepting. I am sorry, but I don't know references for exercises by heart, and I haven't used these concepts much yet. But you may check in relevant books, e.g. the book by Meyn and Tweedie, or I suppose that in Kallenberg's "Foundations of Modern Probability" there might be exercises, at least there will be more references on this topic. Last but not least, Grimmett and Stirzaker might provide exercises. –  Pascal Maillard Apr 28 '12 at 20:10

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