  We investigate the translation lengths of group elements that arise in random
walks on weakly hyperbolic groups. In particular, without any moment condition,
we prove that non-elementary random walks exhibit at least linear growth of
translation lengths. As a corollary, almost every random walk on mapping class
groups eventually becomes pseudo-Anosov and almost every random walk on
$\mathrm{Out}(F_n)$ eventually becomes fully irreducible. If the underlying
measure further has finite first moment, then the growth rate of translation
lengths is equal to the drift, the escape rate of the random walk.
  We then apply our technique to investigate the random walks induced by the
action of mapping class groups on Teichm\"uller spaces. In particular, we prove
the spectral theorem under finite first moment condition, generalizing a result
of Dahmani and Horbez.
