  We study random walks on the isometry group of a Gromov hyperbolic space or
Teichm\"uller space. We prove that the translation lengths of random isometries
satisfy a central limit theorem if and only if the random walk has finite
second moment. While doing this, we recover the central limit theorem of
Benoist and Quint for the displacement of a reference point and establish its
converse. Also discussed are the corresponding laws of the iterated logarithm.
Finally, we prove sublinear geodesic tracking by random walks with finite
$(1/2)$-th moment and logarithmic tracking by random walks with finite
exponential moment.
