Adaptive control of Markov chains under the weak accessibility

The author considers the adaptive control of Markov chains under the weak accessibility condition with the objective of minimizing the learning loss. First, it is shown that, by using a stationary randomized control scheme, the maximum likelihood estimate of the unknown parameter converges exponentially fast to its true value. Then a certainty equivalence control with a forcing type scheme is constructed with alternative phases of forcing and certainty equivalence control. The stationary randomized control scheme for forcing is used in such a way that by cutting and pasting the resulting observations a single Markov chain is obtained. This in turn allows the rate of forcing to be chosen appropriately, giving a learning loss of O(f(n)log n) for any function f (n)→∞ as n→∞