  We consider zero-sum repeated games with incomplete information on both
sides, where the states privately observed by each player follow independent
Markov chains. It generalizes the model, introduced by Aumann and Maschler in
the sixties and solved by Mertens and Zamir in the seventies, where the private
states of the players were fixed. It also includes the model introduced in
Renault \cite{R2006}, of Markov chain repeated games with lack of information
on one side, where only one player privately observes the sequence of states.
We prove here that the limit value exists, and we obtain a characterization via
the Mertens-Zamir system, where the "non revealing value function" plugged in
the system is now defined as the limit value of an auxiliary "non revealing"
dynamic game. This non revealing game is defined by restricting the players not
to reveal any information on the {\it limit behavior} of their own Markov
chain, as in Renault 2006. There are two key technical difficulties in the
proof: 1) proving regularity, in the sense of equicontinuity, of the $T$-stage
non revealing value functions, and 2) constructing strategies by blocks in
order to link the values of the non revealing games with the original values.
