  We consider a continuous-time random walk which is the generalization, by
means of the introduction of waiting periods on sites, of the one-dimensional
nonhomogeneous random walk with a position-dependent drift known in the
mathematical literature as Gillis random walk. This modified stochastic process
allows to significantly change local, non-local and transport properties in the
presence of heavy-tailed waiting-time distributions lacking the first moment:
we provide here exact results concerning hitting times, first-time events,
survival probabilities, occupation times, the moments spectrum and the
statistics of records. Specifically, normal diffusion gives way to subdiffusion
and we are witnessing the breaking of ergodicity. Furthermore we also test our
theoretical predictions with numerical simulations.
