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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?

Alexis
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J. Zeitouni
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  • Yes, data augmentation is a type of Gibbs sampler. While $X_t\sim f$ for all $t$'s when $X_0\sim f$, this does not imply they are iid from $f$. They are marginally from $f$, not jointly (iid) from $f$ – Xi'an May 14 '23 at 17:11
  • See as an illustration the case of the AR(1) Markov chain. – Xi'an May 14 '23 at 19:29
  • Thanks Xi'an! I've been following you and your blog for a while. By this reasoning, the Markov Chain that produces $X_{t+1}=X_{t}$ would also satisfy this condition? – J. Zeitouni May 15 '23 at 06:49
  • The constant Markov chain is the endless repetition of one single value, which is a realisation of the stationary distribution in the case it is generated that way. – Xi'an May 15 '23 at 10:52

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