arXiv:2611.14088v1 Announce Type: cross Abstract: We study discrete-time stochastic control problems where the noise driving the state dynamics is a stationary infinite-order chain rather than an i.i.d. sequence. We formulate the problem as a controlled Markov process by enlarging the state space to include the posterior distribution of the noise given the state-action history. Assuming that the sensitivity of the noise transition kernel to past noise values decays exponentially, we show that the expected total variation distance between the full-history and finite-memory noise filters decays at rate $\sqrt{N}\rho^{\sqrt{N}}$. We use this result to construct finite-memory control models and establish explicit near-optimality bounds for their policies. Finally, under suitable ergodicity conditions, we show that Q-learning converges to the optimal solution of the approximate model, so the same performance bounds apply to the learned policies.