  We prove central limit theorems for the random walks on either the mapping
class group of a closed, connected, orientable, hyperbolic surface, or on
$\text{Out}(F_N)$, each time under a finite second moment condition on the
measure (either with respect to the Teichm\"uller metric, or with respect to
the Lipschitz metric on outer space). In the mapping class group case, this
describes the spread of the hyperbolic length of a simple closed curve on the
surface after applying a random product of mapping classes. In the case of
$\text{Out}(F_N)$, this describes the spread of the length of primitive
conjugacy classes in $F_N$ under random products of outer automorphisms. Both
results are based on a general criterion for establishing a central limit
theorem for the Busemann cocycle on the horoboundary of a metric space, applied
to either the Teichm\"uller space of the surface, or to Culler--Vogtmann's
outer space.
