Question

Bradley (2005, Section 3.1) states:

Theorem 3.1 Suppose $X := (X_k; k \in Z)$ is a strictly stationary, finite state Markov chain. Then the following five statements are equivalent:

1. X is irreducible and aperiodic.
2. X is mixing (in the ergodic-theoretic sense).
3. $\alpha(n) \rightarrow 0$ as $n \rightarrow \infty$.
4. $\psi(n) \rightarrow 0$ as $n \rightarrow \infty$.
5. $\rho^*(n) \rightarrow 0$ as $n \rightarrow \infty$.

Does the theorem hold if $X$ is merely a semi Markov chain? What conditions might apply?

Likewise does the theorem hold if $X$ is merely a regenerative process? Again conditions might apply? In particular, is it enough that $X$ is aperiodic and positive recurrent?

It seems to me that with both semi Markov chains and regenerative processes, the process has 'resetting' properties that could enable strong mixing?

Why I am asking

Let $Z_t$ be an alternating renewal process that is aperiodic and positive recurrent. Choose $\delta t > 0$, put $t_k = k \cdot \delta t$ for each integer $k \geq 0$, and then put $X_k = Z_{t_k}$ for each $k$.

I am studying a sequence $S_n = \sum_{k=1}^n a_{nk} X_k$ where $a_{nk}$ are numeric constants. My hypothesis (supported by experiments) is that $S_n / \text{Var}({S_n})$ converges in distribution to $\mathcal{N}(0,1)$ as $n \rightarrow \infty$.

To this end, I am seeking to invoke Peligrad (1996), Corollary 2.1. Put $\xi_{ni} = a_{ni} X_i$ for each $n$ and $1 \leq i \leq n$. Define $$\bar{\rho}^{*}_{nk} = \sup_{k}(\sigma(\xi_{ni}, i \in T), \sigma(\xi_{nj}, j \in S))$$ where $T,S \subset 1\dots n$ are nonempty and dist$(T,S) \geq k$, and $$\bar{\rho}_{k}^{*} = \sup_{n}\bar{\rho}^{*}_{nk}$$ Peligrad's result requires that $\{X_k\}$ is strongly mixing and that $\lim_{k\rightarrow\infty} \bar{\rho}_{k}^{*} < 1$ (among other conditions).

I can establish that $\{X_k\}$ is strongly mixing ($\alpha$-mixing) by noting that $Z_t$ is aperiodic, positive recurrent, and regenerative, and thereby invoking Glynn (1982, Theorem 6.3.i). But I am having trouble establishing that the condition on $\bar{\rho}_{k}^{*}$ holds in my system.

References

Bradley, Richard C. (2005), Basic Properties of Strong Mixing Conditions. A Survey and Some Open Questions. Probability Surveys 2, 107-144. doi: 10.1214/154957805100000104

Glynn, Peter W. (1982), Some New Results in Regenerative Process Theory. Technical Report 60, July. DTIC: ADA119153

Peligrad, Magda (1996), On the Asymptotic Normality of Sequences of Weak Dependent Random Variables, J Theoretical Probability 9(3), 703-715.