We study the asymptotic behaviour of occupation times of a transient random walk (RW) in a quenched random environment (RE) on a strip in a subdiffusive regime. The asymptotic behaviour of hitting times, which is a more traditional object of study, is exactly the same. As a particular case, we solve a long standing problem of describing the asymptotic behaviour of a RW with bounded jumps on a one-dimensional lattice. Our technique results from the development of ideas from our previous work (Dolgopyat and Goldsheid 2012 Commun. Math. Phys. 315 241–77) on the simple RWs in RE and those used in Bolthausen and Goldsheid (2000 Commun. Math. Phys. 214 429–47; 2008 Commun. Math. Phys. 278 253–88) and Goldsheid (2008 Probab. Theory Relat. Fields 141 471–511) for the study of random walks on a strip.
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