General discounting versus average reward

Consider an agent interacting with an environment in cycles. In every interaction cycle the agent is rewarded for its performance. We compare the average reward U from cycle I to m (average value) with the future discounted reward V from cycle k to infinity (discounted value). We consider essentially arbitrary (non-geometric) discount sequences and arbitrary reward sequences (non-MDP environments). We show that asymptotically U for m -> infinity and V for k -> infinity are equal, provided both limits exist. Further, if the effective horizon grows linearly with k or faster, then the existence of the limit of U implies that the limit of V exists. Conversely, if the effective horizon grows linearly with k or slower, then existence of the limit of V implies that the limit of U exists.