  Let G be a countable group which acts by isometries on a separable, but not
necessarily proper, Gromov hyperbolic space X. We say the action of G is weakly
hyperbolic if G contains two independent hyperbolic isometries. We show that a
random walk on such G converges to the Gromov boundary almost surely. We apply
the convergence result to show linear progress and linear growth of translation
length, without any assumptions on the moments of the random walk. If the
action is acylindrical, and the random walk has finite entropy and finite
logarithmic moment, we show that the Gromov boundary with the hitting measure
is the Poisson boundary.
