## Markov chains 101

**A**n interesting conundrum from Arnaud Guillin: can you find an MCMC setting where there are two Markov chains and such that the pair is *not* a Markov chain. The standard setting is the opposite when a pair is Markov but at least one of the components is not… Data augmentation being an exception. (Arnaud has a normal example available that is unrelated with MCMC samplers.)

November 19, 2009 at 11:13 pm

Antoine Dreyer also has a simple alternative: take two iid sequences and and define

then both and are Markov chains, but is not a (first order) Markov chain.

November 19, 2009 at 5:53 pm

Nice, Randal!

I was thinking of a coupling strategy on a finite set, with and merging when entering a certain atom together. Marginaly, they are both Markov while jointly, they are not: must remember the last time it entered together (or not)….

November 19, 2009 at 5:24 pm

randie star m’en a donne un autre…: je te mets son mail

“Prends un noyau Q ayant densité q par rapport disons à Lebesgue. Tu construis . Tu poses . Tu poses et tu consideres

c’est à dire la loi conditionnelle à la valeur precedente et suivante.

On a clairement que sont marginalement des chaines de Markov. Mais si tu ecris la loi jointe de , tu vas voir que ca ne s’écrit pas en produit de fonctions de et , ca suffit pour dire que ce n’est pas une chaine de Markov. “