I am reading chapter 10 of the Handbook of Markov Chain Monte Carlo (Chapman and Hall/CRC, 2011), by James P. Hobert on the Data Augmentation algorithm, which honestly just looks like a Gibbs sampler to me.
In section 10.2.4 (page 268) on the Central Limit Theorem, they calculate the covariance between $X_1$ and $X_0$, assuming that $X_0$ is sampled from the invariant distribution $f_X$, which implied that $X_1$ is also sampled from $f_X$. If this is the case, shouldn't they both independent? In what sense are they both sampled from an invariant distribution if they're not independent samples?
