A decision theoretical characterization of weak ergodicity

Summary The relation between the ergodic coefficient and deficiency relative to the least informative experiment is investigated. The result is applied to nonhomogeneous Markov chains (NMC's). Our main result can be described as follows: Given an NMC, define the experiments _n ^(j) for n1 consisting in observing the (n+j)-th state of the chain, the j-th state being the unknown parameter. Then the chain is weakly ergodic if and only if for any j, _n ^(j) converges as n (with respect to deficiencies) to the least informative experiment. It is finally shown that in the homogeneous case, the rate of convergence is always exponential.