  Doubly robust methods hold considerable promise for off-policy evaluation in
Markov decision processes (MDPs) under sequential ignorability: They have been
shown to converge as $1/\sqrt{T}$ with the horizon $T$, to be statistically
efficient in large samples, and to allow for modular implementation where
preliminary estimation tasks can be executed using standard reinforcement
learning techniques. Existing results, however, make heavy use of a strong
distributional overlap assumption whereby the stationary distributions of the
target policy and the data-collection policy are within a bounded factor of
each other -- and this assumption is typically only credible when the state
space of the MDP is bounded. In this paper, we re-visit the task of off-policy
evaluation in MDPs under a weaker notion of distributional overlap, and
introduce a class of truncated doubly robust (TDR) estimators which we find to
perform well in this setting. When the distribution ratio of the target and
data-collection policies is square-integrable (but not necessarily bounded),
our approach recovers the large-sample behavior previously established under
strong distributional overlap. When this ratio is not square-integrable, TDR is
still consistent but with a slower-than-$1/\sqrt{T}$; furthermore, this rate of
convergence is minimax over a class of MDPs defined only using mixing
conditions. We validate our approach numerically and find that, in our
experiments, appropriate truncation plays a major role in enabling accurate
off-policy evaluation when strong distributional overlap does not hold.
