  A random walk $w_n$ on a separable, geodesic hyperbolic metric space $X$
converges to the boundary $\partial X$ with probability one when the step
distribution supports two independent loxodromics. In particular, the random
walk makes positive linear progress. Progress is known to be linear with
exponential decay when (1) the step distribution has exponential tail and (2)
the action on $X$ is acylindrical. We extend exponential decay to the
non-acylindrical case.
