I came across a curio today while looking at recent postings on arXiv, namely a different proof of the convergence of the Data Augmentation algorithm, more than twenty years after it was proposed by Martin Tanner and Wing Hung Wong in a 1987 JASA paper… The convergence under the positivity condition is of course a direct consequence of the ergodic theorem, as shown for instance by Tierney (1994), but the note by Yaming Yu uses instead a Kullback divergence
and shows as Liu, Wong and Kong do for the variance (Biometrika, 1994) that this divergence is monotonically decreasing in t. The proof is interesting in that only functional (i.e., non-ergodic) arguments are used, even though I am a wee surprised at IEEE Transactions on Information Theory publishing this type of arcane mathematics… Note that the above divergence is the “wrong” one in that it measures the divergence from 
