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Let $P$ be a stochastic kernel on a measurable space $(\mathsf X,\mathfrak B(\mathsf X))$. The kernel $P$ is called $\varphi$-irreducible if for a positive measure $\varphi$ and for all measurable sets $A$ it holds that: $$ \tag{1}\varphi(A)>0 \quad \Rightarrow \quad \sum\limits_{n\geq 1}P^n(x,A) >0 \quad \forall x\in \mathsf X. $$

One of the statements of Proposition 4.2.2 (p. 90 here) is that if $P$ is $\varphi$-irreducible, then it holds that $P$ is $\psi$-irreducible where the measure $\psi$ is given by $$ \psi(A) = \int\limits_\mathsf X \sum_{n\geq 0}2^{-(n+1)}P^n(x,A)\varphi(\mathrm dx). $$ The definition of $\psi$ is not that important for my question, though.

In the first part of the proof, also page 90, the following is stated

To see (i), observe that when $\psi(A)>0$, then [...] $$ \tag{2} \left\{y:\sum\limits_{n\geq 1}P^n(y,A)>0\right\} = \mathsf X. $$

This fact is further elaborated and used to show the $\psi$-irreducibility of $P$. It seems to me, however, that the cited part explicitly implies irreducibility as it is equivalent to $(1)$. I guess, I am missing something - otherwise it is a cyclic argument. Also, I don't know how to show $(2)$.

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How is Markov chain splitting technique useful for inferring ergodicity of a Markov Chain?Assume that I am working with general state space (uncountable say $R^{N}$ but time is discrete. I want to show that the Markov Process is ergodic. I guess that it suffices to show that it is Harris recurrent. To show Harris recurrence I guess that it suffices to show there exists an atom (obtained via splitting the chain after using minorization criteria) the return time (or hitting time) to which has finite mean. –  user24367 Jun 18 '13 at 13:45
    
@user17523: I think, it is better if you ask a separate question. I'm not an expert in the book by Meyn and Tweedie, and I didn't read much of their chapters on ergodicity. I remember splitting technique, but I haven't reached the part where it's used. –  Ilya Jun 19 '13 at 7:47
    
Hi, I already posted it as separate question, but also posted here since I saw a similar topic being discussed here. –  user24367 Jun 19 '13 at 18:32

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