Simulation of the mean position of a Reflected Random Walk (RRW)
exhibits non-standard behavior, even for light-tailed increment
distributions with negative drift. The Large Deviation Principle
(LDP) holds for deviations below the mean, but for deviations at
the usual speed, above the mean the rate function is null. This
talk takes a deeper look at the latter phenomenon. Conditional on
a large sample mean, a complete sample path LDP analysis is described
that helps to explain why simulating a RRW is hard. This gives rise
to non-convex rate function and elegant concave most likely paths.
As a riposte, given time, we shall show that, under restrictive
assumptions, estimating the tail behavior of the stationary
distribution of a RRW is relatively easy and conjecture that this
is true in substantially greater generality.