  We investigate the translation lengths of group elements that arise in random
walks on the isometry groups of Gromov hyperbolic spaces. 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{\"u}ller spaces. In particular, we
prove the spectral theorem under finite first moment condition, generalizing a
result of Dahmani and Horbez.
