# $\psi$-irreducibility of m-skeletons.

In Proposition 5.4.5 of Meyn and Tweedie's Markov Chains and Stochastic Stability, it is said that if a chain $\Phi$ is $\psi$-irreducible and aperiodic, then every $m$-skeleton of it is also $\psi$-irreducible and aperiodic. The authors did not give an explicit proof of this.

It is easy to show aperiodicity using induction, but I had a hard time to figure out the proof for $\psi$-irreducibility. Is induction and contradiction a right direction to go? In general, if there is no aperiodicity of the chain, is it also true that all the $m$-skeletons are $\psi$-irreducible?

For any $$C \in B^{+}(X)$$, w.o.l.g. (see Theorem 5.2.2), we can assume that $$C$$ is a $$\nu_M$$ small set with $$\nu_M(C) > 0$$ . Since, $$\Phi$$ is aperiodic (and because this property does not depend on the small set as shown by the proof of Theorem 5.4.4), $$g.c.d.(E_C)=1$$. By Lemma D.7.4, there exists $$k_0$$ s.t. for all $$k \geq k_0$$, $$C$$ is $$\nu_k$$ small with $$\nu_k = \delta_k \nu_M$$ and with $$\delta_k >0$$.
For $$x \in X$$, since $$\Phi$$ is $$\psi$$-irreducible, there is some $$r$$ s.t. $$P^r(x, C) >0$$. Hence, by Chapman-Kolmogorov, for any $$k \geq k_0$$ we get: $$P^{r+k}(x, C) \geq \int_C P^r(x, dy) P^k(y, C) \geq P^r(x, C) \delta_k \nu_M(C) >0 \quad .$$ Therefore, for any $$m$$, there exist $$i$$ s.t. $$P^{i\cdot m}(x, C) >0$$, which yields the result.