  In this paper, for POMDPs, we provide the convergence of a Q learning
algorithm for control policies using a finite history of past observations and
control actions, and, consequentially, we establish near optimality of such
limit Q functions under explicit filter stability conditions. We present
explicit error bounds relating the approximation error to the length of the
finite history window. We establish the convergence of such Q-learning
iterations under mild ergodicity assumptions on the state process during the
exploration phase. We further show that the limit fixed point equation gives an
optimal solution for an approximate belief-MDP. We then provide bounds on the
performance of the policy obtained using the limit Q values compared to the
performance of the optimal policy for the POMDP, where we also present explicit
conditions using recent results on filter stability in controlled POMDPs. While
there exist many experimental results, (i) the rigorous asymptotic convergence
(to an approximate MDP value function) for such finite-memory Q-learning
algorithms, and (ii) the near optimality with an explicit rate of convergence
(in the memory size) are results that are new to the literature, to our
knowledge.
