2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/221368We consider a one-dimensional recurrent random walk in random environment (RWRE). We show that the - suitably centered - empirical distributions of the RWRE converge weakly to a certain limit law which describes the stationary distribution of a random walk in an infinite valley. The construction of the infinite valley goes back to Golosov. As a consequence, we show weak convergence for both the maximal local time and the self-intersection local time of the RWRE and also determine the exact constant in the almost sure upper limit of the maximal local time.17 pages, 1 figureProbability60K37, 60J50, 60J55, 60F10text