Let $\hat{\mu_t}(f)=\frac{1}{t} \sum_{i=1}^{t} f(X_i)$ is an estimator for a finite, irreducible Markov chain $\{X_t\}$ with its stationary distribution $\pi$. In addition, assume the estimator is unbiased.
Then, let's consider the asymptotic variance, which is given by
$\sigma^2 (f) = \lim_{t \rightarrow \infty} t \cdot Var(\hat{\mu_t})=\lim_{t \rightarrow \infty} \frac{1}{t} E \{ [\sum_{i=1}^{t} f(X_i) - E_{\pi}(f)]^2 \}$.
Moreover, assume that we have another Markov chain $\{X_t^{'}\}$ with the same stationary distribution $\pi$ as one for $\{X_t\}$, but have different transition matrix and the estimator $\hat{\mu_t}^{'}(f)=\frac{1}{t} \sum_{i=1}^{t} f(X_i^{'})$
In this case, can the asymptotic variances for $\hat{\mu_t}^{'}$ and $\hat{\mu_t}$ be different?
I think it should be same due to 1) the same stationary distribution and 2) the same form of the estimator function, but some papers regard them as different (in my understanding).