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

Linear growth of translation lengths of random isometries on Gromov hyperbolic spaces and Teichm\"uller spaces