  In the proof of the invariance principle for locally perturbed periodic
Lorentz process with finite horizon, a lot of delicate results were needed
concerning the recurrence properties of its unperturbed version. These were
analogous to the similar properties of Simple Symmetric Random Walk. However,
in the case of Lorentz process with infinite horizon, the analogous results for
the corresponding random walk are not known, either. In this paper, these
properties are ascertained for the appropriate random walk (this happens to be
in the non normal domain of attraction of the normal law). As a tool, an
estimation of the remainder term in the local limit theorem for the
corresponding random walk is computed.
