  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 1 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 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.
